A Calculus Refresher

Limits

This is a very condensed and simplified version of basic calculus, which is a prerequisite for the courses you take during your honours year. It is not comprehensive, and absolutely not intended to be a substitute for a proper course in differential and integral calculus. You are strongly encouraged to solve the included Exercises to reinforce the ideas.

[Notes developed by Daniela Castro-Camilo]

Limits

A limit is the value that a function (or sequence) “approaches” as the input (or index) “approaches” some other value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

Limit of a function

Let \(f(x)\) be a real-valued function and let \(a\in\mathbb{R}\). We say that the limit of \(f\) as \(x\) approaches \(a\) is \(L\) if for every \(\epsilon>0\) there exist \(\delta>0\) such that \[|x-a|< \delta\implies |f(x)-L|<\epsilon.\] We write \(L = \lim_{x\to a}f(x)\).

The definition states that for any number \(\epsilon>0\) we pick, we can go to our plot in Figure 1 and sketch two horizontal lines at \(L-\epsilon\) and \(L+\epsilon\). Then, it is possible to find another real number \(\delta>0\) (which we need to determine) that allows us to add in two vertical lines to our plot at \(a-\delta\) and \(a+\delta\). If we take any \(x\) close enough to \(a\) (and by close enough we mean in the pink region, i.e., between \(a-\delta\) and \(a+\delta\)), then the image of \(x\) under \(f\), i.e., \(f(x)\) will be within \(L-\epsilon\) and \(L+\epsilon\), i.e., it will be close enough to \(L\). In other words, by making \(x\) sufficiently close to \(\boldsymbol{a}\), you can set \(\boldsymbol{f(x)}\) as close as you want to \(\boldsymbol{L}\).

Figure 1. Illustration of the limit of a function.

Limit of a sequence

A sequence \(x_n\in\mathbb{R}\), \(n\in\mathbb{N}\), is said to converge to \(a\in\mathbb{R}\) if for all \(\epsilon>0\) there is \(N_\epsilon\in\mathbb{N}\) such that for all \(n>N_\epsilon\), \(x_n\) does not deviate from \(a\) by more than \(\epsilon\), i.e., if \[\forall \epsilon>0\hskip0.2cm \exists N_\epsilon\in\mathbb{N}: n>N_\epsilon\implies |x_n-a|< \epsilon.\]

We write \(a=\lim_{n\to\infty}x_n\).

Figure 2. Illustration of the limit of a sequence.

Properties of the limit of a function

The limit of \(f\) at \(a\) is \(L\) if and only if the limit from above (right) and below (left) of \(f\) at \(a\) are equal to \(L\), i.e., \[\lim_{x\to a^+}f(x) = \lim_{x\to a^-}f(x) = L.\]
Continuity. The function \(f\) is continuous at \(a\) if and only if \(\lim_{x\to a}f(x) = f(a).\)
The algebraic limit theorem. If \(f\) and \(g\) are real-valued functions, then taking the limit of an operation on \(f(x)\) and \(g(x)\) is under certain conditions compatible with the algebraic operations and exponentiation. The main condition needed for applying the following rules is that the limits on the right-hand sides of the equations exist.

3.1. \(\lim_{x\to a} [f(x) \pm g(x)] = \lim_{x\to a} f(x) \pm \lim_{x\to a}g(x).\)

3.2. \(\lim_{x\to a}f(x)\cdot g(x) = \lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x).\)

3.3. \(\lim_{x\to a}f(x)/g(x) = \lim_{x\to a}f(x)/\lim_{x\to a}g(x),\) provided \(\lim_{x\to a}g(x)\neq0.\)

3.4. \(\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\lim_{x\to a}g(x)}\), if the base is positive, or zero while the exponent comes out positive (but finite).
L’Hôpital’s rule. Let \(f(x)\) and \(g(x)\) be two functions defined over an open interval \(I\) containing the desired limit point \(a\). Assume that
1. \(\lim_{x\to a}f(x) = \lim_{x\to a}g(x) = 0\), or \(\lim_{x\to a}f(x) = \pm \lim_{x\to a}g(x) = \pm\infty\).
2. \(f\) and \(g\) are differentiable over \(I\setminus\{a\}\).
3. \(g'(x)\neq 0\) for all \(x\in I\setminus\{a\}\).
4. \(\lim_{x\to a}f'(x)/g'(x)\) exists.
Then \[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}.\] Normally, the first condition is the most important one.

Properties of the limit of a sequence

The properties of limits of function also apply to limits of sequences. For the sake of completeless we list them here.

\(\lim_{n\to \infty} [a_n \pm b_n] = \lim_{n\to \infty} a_n \pm \lim_{n\to \infty}b_n.\)
\(\lim_{n\to \infty} a_n \cdot b_n = \lim_{n\to \infty} a_n \cdot \lim_{n\to \infty}b_n.\)
\(\lim_{n\to \infty} a_n/b_n = \lim_{n\to \infty} a_n/\lim_{n\to \infty}b_n\), provided \(\lim_{n\to \infty}b_n\neq 0.\)
\(\lim_{n\to \infty} a_n^p = [\lim_{n\to \infty} a_n]^p.\)
If \(a_n\leq b_n\) for all \(n\) greater than some \(N\), then \(\lim_{n\to \infty} a_n\leq \lim_{n\to \infty} b_n\).
Squeeze theorem. If \(a_n\leq c_n\leq b_n\) for all \(n>N\), and \(\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n = L\), then \(\lim_{n\to\infty} c_n = L\).
If a sequence is bounded and monotonic, then it is convergent.
A sequence is convergent if and only if every subsequence is convergent.
If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.

Exercises

Compute \(\displaystyle \lim_{x\to\pi/4}\frac{\sin x - \cos x}{\cos(2x)}\)
Solution
\(\displaystyle-\sqrt{2}/2\)
Show that \(\displaystyle \lim_{x\to 0}x^4\cos(2/x) = 0\)
Hint
Use the Squeeze Theorem.
Find the value \(k\) for which the following limit exists: \(\displaystyle \lim_{x\to 3}\frac{4x^2+kx+7k-6}{2x^2-5x-3}\).
Solution
\(k=3\)
Let \[\displaystyle a_k =\frac{1}{k2^k}\quad \text{and}\quad b_k=\frac{k}{2^k}.\] Find the limits of the sequences \(a_{k+1}/a_k\) and \(b_{k+1}/b_k\).
Solution
\(\displaystyle \lim_{k\to\infty}a_{k+1}/a_k = \lim_{k\to\infty}b_{k+1}/b_k = 1/2\)
Determine the limit of the sequence whose \(n\)-th term is \(\displaystyle x_n=\sqrt{n^4+n^2}-n^2\).
Solution
\(\lim_{n\to\infty}x_n =1/2.\)

Derivatives

The derivative of a function \(f:\mathbb{R}\to \mathbb{R}\) measures the sensitivity to change of the function value \(f(x)\) with respect to a change in its argument, \(x\). The derivative of a function \(f(x)\) of a single variable at \(x_0\), when it exists, is the slope of the tangent line to the graph of the function at \(x_0\). The tangent line is the best linear approximation of \(f(x)\) near that \(x_0\). For this reason, the derivative is often described as the “instantaneous rate of change”.

Formal definition

If a curve \(C\) has equation \(y=f(x)\) and we want to find the tangent line to \(C\) at the point \(P(a,f(a))\), then we consider a nearby point \(Q(x,f(x))\), where \(x\neq a\), and compute the slope of the secant line \(PQ\): \[m_{PQ} = \frac{f(x)-f(a)}{x-a}.\] Then we let \(Q\) approach \(P\) along the curve \(C\) by letting \(x\) approach \(a\). If \(m_{PQ}\) approaches a number \(m\), then we define the tangent \(t\) to be the line through \(P\) with slope \(m\). See Figure 3.

Definition. The tangent line to the curve \(y=f(x)\) at the point \(P(a,f(a))\) is the line through \(P\) with slope \[m = \lim_{x\to a}\frac{f(x)-f(a)}{x-a},\] provided that this limit exists.

Figure 3. Point-slope form for a line through the point \((x_1,y_1)\) with slope \(m\): \(y-y_1=m(x-x_1)\). Figure taken from Stewart (2008).

Definition. The derivative of a function \(\boldsymbol{f}\) at a number \(\boldsymbol{a}\), denoted by \(f'(a)\), is \[f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h},\] provided that this limit exists. If we write \(x=a+h\), then we have \(h=x-a\) and \(h\) approaches 0 if and only if \(x\) approaches \(a\). Therefore, an equivalent way of stating the definition of he derivative, as we saw in finding tangent lines, is \[f'(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}.\]

Definition. A function \(f\) is differentiable at a if \(f'(a)\) exists. It is differentiable on an open interval (a,b) if it differentiable at every number in the interval.

Theorem. If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

Rules

Differentiation is linear. For any functions \(f\) and \(g\) and any real numbers \(a\) and \(b\), the derivative of the function \(h(x)=af(x)+bg(x)\) with respect to \(x\) is

\[h'(x)=af'(x)+bg'(x).\]

The product rule. The derivative of the function \(h(x) = f(x) g(x)\) with respect to \(x\) is

\[h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).\]

Chain rule. The derivative of the function \(h(x)=f(g(x))\) is

\[h'(x)=f'(g(x))\cdot g'(x).\]

The inverse function rule. If the function \(f\) has an inverse function \(g\), meaning that \(g(f(x))=x\) and \(f(g(y))=y\), then

\[g'=\frac{1}{f'\circ g}.\]

The polynomial or elementary power rule. If \(f(x) = x^r\), for any real number \(r\neq 0\), then

\[f'(x)=rx^{r-1}.\]

Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.

The reciprocal rule. The derivative of \(h(x)=\frac{1}{f(x)}\) for any (nonvanishing) function \(f\) is:

\[h'(x)=-\frac{f'(x)}{(f(x))^{2}},\quad \text{wherever } f \text{ is non-zero.}\]

The quotient rule. If \(f\) and \(g\) are functions, then:

\[\left(\frac{f}{g}\right)'=\frac{f'g-g'f}{g^{2}}\quad \text{wherever } g \text{ is nonzero.}\] This can be derived from the product rule and the reciprocal rule.

Generalized power rule. For any functions \(f\) and \(g\),

\[(f^g)'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\] wherever both sides are well defined.

Implicit differentiation

Functions can be defined implicitly by a relation between, say, \(x\) and \(y\). For example \[\begin{equation*} x^2+y^2=25 \end{equation*}\] or \[\begin{equation} x^3+y^3=6xy (\#eq:descartes) \end{equation}\]

In some cases it is possible to solve such an equation for \(y\) as an explicit function of \(x\). Nonetheless, equations such as @ref(eq:descartes) which represents a curve called the folium of Descartes (see Figure 4) cannot be described explicitly. Fortunately, we don’t need to solve an equation for \(y\) in terms of \(x\) in order to find the derivative of \(y\). Instead, we differentiate both sides of the equation with respect to \(x\) and then solve the result equation for \(y'\). This is called the method of implicit differentiation.

Figure 4. The folium of Descartes. Figure taken from Stewart (2008).

Example. Differentiating both sides of \(x^3+y^3=6xy\) with respect to \(x\) gives \[3x^2+3y^2y' = 6xy' + 6y\qquad\text{or}\qquad x^2+y^2y' = 2xy' + 2y.\]

Solving for \(y'\) we get \(y'=(2y-x^2)/(y^2-2x).\)

Uses of differentiation

Increasing and Decreasing Functions

An increasing function is a function where: if \(x_1 > x_2\), then \(f(x_1) > f(x_2)\) , so as \(x\) increases, \(f(x)\) increases. A decreasing function is a function which decreases as \(x\) increases. Of course, a function may be increasing in some places and decreasing in others. A point where a function changes from an increasing to a decreasing function or visa-versa is known as a turning point. A turning point is a type of stationary point (see below). We can use differentiation to determine if a function is increasing or decreasing:

A function is increasing if its derivative is always positive. A function is decreasing if its derivative is always negative.

Examples

\(y = -x\) has derivative -1 which is always negative and so \(-x\) is decreasing.
\(y = x^2\) has derivative \(2x\), which is negative when \(x<0\) and positive when \(x>0\). Hence \(x^2\) is decreasing for \(x<0\) and increasing for \(x>0\).

Stationary Points

Stationary points are points on a graph where the gradient is zero. The gradient is defined for a scalar-valued differentiable function \(f(x_1,\ldots,x_n)\) of several variables as the vector whose components are the partial derivatives. If \(n=1\), then the gradient is just the derivative of \(f\). There are three types of stationary points: maximum, minimum and points of inflection. The three are illustrated in Figure 5.

Figure 5. The three types of stationary points.

Maximum, Minimum or Point of Inflection?

At all the stationary points, the gradient is the same (\(=0\)) but it is often necessary to know whether you have found a maximum point, a minimum point or a point of inflection. Therefore the gradient at either side of the stationary point needs to be looked at (alternatively, we can use the second derivative, see below).

At maximum points, the gradient is positive just before the maximum, it is zero at the maximum and it is negative just after the maximum. At minimum points, the gradient is negative, zero then positive. Finally at points of inflexion, the gradient can be positive, zero, positive or negative, zero, negative.

Second derivative

The second derivative is what you get when you differentiate the derivative. The second derivative can be used as an easier way of determining the nature of stationary points (whether they are maximum points, minimum points or points of inflection).

A stationary point \(x_0\) on a curve occurs when \(f'(x_0)=0\). Once we have established where there is a stationary point, the type of stationary point (maximum, minimum or point of inflexion) can be determined using the second derivative as follows:

If \(f''(x_0)>0\), then \(f(x_0)\) is a minimum point.
If \(f''(x_0)<0\), then \(f(x_0)\) is a maximum point.
If \(f''(x_0)=0\), then \(f(x_0)\) could be a maximum, minimum or point of inflexion.

Exercises

Find the derivative of \(\displaystyle y = \sqrt{x^3\sqrt{x^5}\sqrt{x^7}}\)
Solution
\(\displaystyle\frac{9x^{7/2}}{2}\)
Find the derivative of \(\displaystyle y = (\sin x)^{\cos x}\)
Solution
\(\displaystyle(\sin x)^{\cos x}\left(\frac{\cos^2 x}{\sin x} - \sin x \ln(\sin x)\right)\)
Find the tangent to the folium of Descartes \(\displaystyle x^3+y^3 = 6xy\) at the point (3,3).
Solution
\(\displaystyle x+y = 6\)
Determine where the function \(\displaystyle h(z) = 6+40`^3-5z^4-4z^5\) is increasing and decreasing.
Solution
Increasing: \(\displaystyle-3<z<0,\quad 0<z<2.\)

Decreasing: \(\displaystyle\infty<z<-3,\quad 2<z<\infty.\)
Determine where, if anywhere, the tangent line to \(\displaystyle f(x) = x^3-5x^2+x\) is parallel to the line \(y=4x+23.\)
Solution
The tangent line is parallel to \(\displaystyle y=4x+23\) at \(\displaystyle x = \frac{5\pm\sqrt{34}}{3}.\)

Integrals

The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative.

Riemann integrals

There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals. Here, we focus on Riemann integrals.

Definition. The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let \([a, b]\) be a closed interval of the real line; then a tagged partition of \([a, b]\) is a finite sequence \[a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\] This partitions the interval \([a, b]\) into n sub-intervals \([x_{i−1}, x_i]\) indexed by \(i\), each of which is “tagged” with a distinguished point \(t_i \in [x_{i−1}, x_i]\). A Riemann sum of a function \(f\) with respect to such a tagged partition is defined as \[{\displaystyle \sum _{i=1}^{n}f(t_{i})\,\Delta _{i}.}\] Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let \(\Delta_i = xi−x_{i−1}\) be the width of sub-interval \(i\); then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, \(\max_{i=1,\ldots,n} \Delta_i\). The Riemann integral of a function \(f\) over the interval \([a, b]\) is equal to \(S\) if: for all \(\epsilon > 0\) there exists \(\delta > 0\) such that, for any tagged partition \([a, b]\) with mesh less than \(\delta\), we have \[{\displaystyle \left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .}\] We write \(S = \int _{a}^{b}f(x)\,dx.\)

Area under a curve

An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral \[\int _{a}^{b}f(x)\,dx\] can be interpreted informally as the signed area of the region in the plane that is bounded by the graph of \(f\), the x-axis and the vertical lines \(x = a\) and \(x = b\). The area above the \(x\)-axis adds to the total and that below the \(x\)-axis subtracts from the total (see Figure 6).

Figure 6. A definite integral of a function can be represented as the signed area of the region bounded by its plot.

Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiation with the concept of integration. There are two parts to the theorem. The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.

First part. Let \(f\) be a continuous real-valued function defined on a closed interval \([a, b]\). Let \(F\) be the function defined, for all \(x \in [a, b]\), by

\[F(x)=\int _{a}^{x}\!f(t)\,dt.\] Then \(F\) is uniformly continuous on \([a, b]\) and differentiable on the open interval \((a, b)\), and

\[F'(x)=f(x), \qquad\text{for all }x \in (a, b).\]

Corollary. If \(f\) is a real-valued continuous function on \([a,b]\) and \(F\) is an antiderivative of \(f\) in \([a,b]\) then

\[\int _{a}^{b}f(t)\,dt=F(b)-F(a).\] The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following part of the theorem.

Second part (the Newton–Leibniz axiom). Let \(f\) be a real-valued function on a closed interval \([a,b]\) and \(F\) an antiderivative of \(f\) in \([a,b]\), i.e., \(F'(x)=f(x).\) If \(f\) is Riemann integrable on \([a,b]\) then

\[\int _{a}^{b}f(x)\,dx=F(b)-F(a).\] The second part is somewhat stronger than the corollary because it does not assume that \(f\) is continuous.

When an antiderivative \(F\) exists, then there are infinitely many antiderivatives for \(f\), obtained by adding an arbitrary constant to \(F\). Also, by the first part of the theorem, antiderivatives of \(f\) always exist when \(f\) is continuous.

Definite and indefinite integrals

The definite integral \[\int_a^bf(x)\,dx\] is a number and represents the area under the curve \(f(x)\) from \(x=a\) to \(x=b\). The indefinite integral (also known as antiderivative) \[\int f(x)\,dx\] is a function and answers the question What function when differentiated gives f(x)?

Example. The function \({\displaystyle F(x)={\tfrac {x^{3}}{3}}}\) is an antiderivative (indefinite integral) of \({\displaystyle f(x)=x^{2}}\) while \(\int_1^2 f(x)\, dx = F(2) - F(1) = {\tfrac {2^{3}}{3}} - {\tfrac {1^{3}}{3}} = {\tfrac {7}{3}}\) is the definite integral of \(f\) between 1 and 2.

Integration techniques

Here are examples of some techniques that are frequently successful when seeking antiderivatives of functions.

Substitution or u-substitution or change of variables. Compute \(\int (2x^{3}+1)^{7}(x^{2})\,dx.\)

Set \(u=2x^{3}+1\). This means \(\frac {du}{dx}=6x^{2}\), or, in differential form \(du=6x^{2}\,dx\). Now

\(\int (2x^{3}+1)^{7}(x^{2})\,dx=\frac {1}{6}\int \underbrace {(2x^{3}+1)^{7}}_{u^{7}}\underbrace {(6x^{2})\,dx}_{du}={\frac {1}{6}}\int u^{7}\,du={\frac {1}{6}}\left({\frac {1}{8}}u^{8}\right)={\frac {1}{48}}(2x^{3}+1)^{8}+C.\)
Trigonometric substitution. Evaluate the integral \({\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx,}.\)

Let \({\displaystyle x=a\sin \theta ,\,dx=a\cos \theta \,d\theta ,\,\theta =\arcsin {\frac {x}{a}},}\) where \({\displaystyle a>0}\) so that \({\displaystyle {\sqrt {a^{2}}}=a}\), and \({\displaystyle -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}}\) by the range of arcsine, so that \({\displaystyle \cos \theta \geq 0}\) and \({\displaystyle {\sqrt {\cos ^{2}\theta }}=\cos \theta }.\) Then,

\[\begin{align*} \int {\sqrt {a^{2}-x^{2}}}\,dx &=\int {\sqrt {a^{2}-a^{2}\sin ^{2}\theta }}\,(a\cos \theta )\,d\theta \\ &=\int {\sqrt {a^{2}(1-\sin ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\ &=\int {\sqrt {a^{2}(\cos ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\ &=\int (a\cos \theta )(a\cos \theta )\,d\theta \\ &=a^{2}\int \cos ^{2}\theta \,d\theta \\ &=a^{2}\int \left({\frac {1+\cos 2\theta }{2}}\right)\,d\theta \\ &={\frac {a^{2}}{2}}\left(\theta +{\frac {\sin(2\theta )}{2}}\right)+C\\ &={\frac {a^{2}}{2}}(\theta +\sin \theta \cos \theta )+C\\ &={\frac {a^{2}}{2}}\left(\arcsin {\frac {x}{a}}+{\frac {x}{a}}{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\right)+C\\ &={\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+{\frac {x{\sqrt {a^{2}-x^{2}}}}{2}}+C. \end{align*}\]
Integration by parts or partial integration. If \({\displaystyle u=u(x)}\) and \({\displaystyle du=u'(x)\,dx}\) while \({\displaystyle v=v(x)}\) and \({\displaystyle dv=v'(x)dx}\), then the integration by parts formula states that \[\begin{align*} \int _{a}^{b}u(x)v'(x)\,dx&={\Big [}u(x)v(x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,dx\\ &=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx. \end{align*}\] More compactly, \[{\displaystyle \int u\,dv\ =\ uv-\int v\,du.}\]

Exercises

\(\displaystyle\int\frac{\sqrt{x^2+1} - \sqrt{x^2-1}}{\sqrt{x^4-1}}\,dx.\)
Solution
\(\displaystyle\ln\Big|x+\sqrt{x^2-1} - \ln\Big|x+\sqrt{x^2-1}\Big|\Big| + C,\quad |x|>1\)
\(\displaystyle\int\frac{\text{sec}^2(x)}{9+\tan^2(x)}\,dx.\)
Solution
\(\displaystyle\frac{1}{3}\arctan\left(\frac{\tan(x)}{3}\right) + C,\quad x\neq \frac{\pi}{2}+k\pi.\)
\(\displaystyle\int\frac{u}{(u^2-4u+13)^2}\,du.\)
Solution
\(\displaystyle\frac{1}{27}\arctan\left(\frac{u-2}{3}\right) + \frac{1}{18}\frac{2u-13}{u^2-4u+13} + C,\quad u\in\mathbb{R}\)
\(\displaystyle\int\frac{1+\cos^2(x)}{1+\cos(2x)}\,dx.\)
Solution
\(\displaystyle\frac{1}{2}\tan(x)+\frac{x}{2}+C,\quad x\neq \frac{\pi}{2}+k\pi\)
\(\displaystyle\int\frac{3x^5-10x^4+21x^3-42x^2+36x-32}{x(x^2+1)(x^2+4)^2}\,dx.\)
Solution
\(\displaystyle-2\ln|x| + 2\arctan(x) + \ln(x^2+4) + \frac{1}{2}\arctan\left(\frac{x}{2}\right) + \frac{1}{x^2+4}+C,\quad x\neq0\)