# Integration Techniques

Some integrals are easy to evaluate, like the first 2 examples below.

What is the indefinite integral of \(x^2\)?

We can write this as \(\int x^2\ dx.\) Using the (integration)

power rule, which states that

\[\int x^n\ dx = \frac{x^{n+1}}{n+1} + C,\] where \(C\) is the constant of integration (which will be used in all of other formulae below), \(x^n =x^2 \implies n=2.\) Therefore,

\[\int x^2\ dx = \frac{x^{2+1}}{2+1} + C = \frac{x^{3}}{3} + C.\ _\square\]

What is the general anti-derivative of \(666x\)?

We can write this as \(\int 666x \ dx.\) Using the

power rule, which states that

\[\int x^n\ dx = \frac{x^{n+1}}{n+1} + C,\] where \(C\) is the constant of integration, and theconstant function rule, which states that

\[\int a\ f(x)\ dx = a \int f(x)\ dx,\] \(x^n =x=x^1 \implies n = 1\) and \(a = 666.\) Therefore,

\[\int 666x\ dx = 666 \cdot \frac{x^{1+1}}{1+1} + C = \frac{666}{2} \cdot x^2 + C = 333x^2 + C.\ _\square\]

#### Contents

## Basic Integration Formulae

Here are fundamental theorems for simple evaluation of integrals.

The

constant rulestates that for an arbitrary constant \(a\)

\[\int a\ dx = ax + C.\]

What is the indefinite integral of \(1\)?

Since \(a=1\) in this case,

\[\int a\ dx = \int 1\ dx = \int dx = 1x + C = x + C.\ _\square\]

\[\]

Theconstant function rule, more formally known as theconstant multiple rule, states that for an arbitrary constant \(a\) and an arbitrary function \(f(x)\) \[\int a\ f(x)\ dx = a \int f(x)\ dx.\]

What is the indefinite integral of \(2x\)?

Since \(a=2\) and \(f(x)=x\) in this case, we have \[\int a\ f(x)\ dx = \int 2x\ dx = 2 \int x\ dx = 2 \frac{x^{1+1}}{1+1} + C = 2 \frac{x^2}2 + C = \frac22x^2 + C = 1x^2 + C = x^2 + C.\ _\square\] Note that we also used the

power rule, which we'll be discussing in a later note.

\[\]

Thesum and difference rulesstate that for arbitrary functions \(f(x)\) and \(g(x)\)

\[\int \big(f(x) \pm g(x)\big)\ dx = \int f(x)\ dx \pm \int g(x)\ dx.\]

What is the indefinite integral of \(2x + 3x^2\)?

Since \(f(x)=2x\) and \(g(x)=3x^2\) in this case, we have \[\int \big(f(x) + g(x)\big)\ dx = \int \big(2x + 3x^2\big)\ dx = \int 2x\ dx + \int 3x^2\ dx = \frac2{1+1}x^{1+1} + \frac3{2+1}x^{2+1} + C = x^2 + x^3 + C.\ _\square\]

\[\]

Thepower rulestates that, for an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n,x \neq 0,\)

\[\int x^n\ dx = \frac{x^{n+1}}{n+1} + C.\]

What is the indefinite integral of \(x\)?

Since \(x^n=x=x^1 \implies n=1\) in this case, we have

\[\int x^n\ dx = \int x\ dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n,x \neq 0,\)

\[\int \frac1{nx} dx = \frac{\ln |nx|}{n} + C.\]

What is the indefinite integral of \(\frac1{2x}\)?

Since \(nx=2x \implies n=2\) in this case, we have \[\int \frac1{nx}\ dx = \int \frac1{2x}\ dx = \frac{\ln |2x|}{2} + C.\ _\square\]

## Trigonometric Integration Formulae

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n \neq 0,\)

\[\int \sin\ nx\ dx = -\frac{1}{n}\ \cos\ nx + C.\]

What is the indefinite integral of \(\sin\ 3x\)?

Since \(nx=3x \implies n=3\) in this case, we have \[\int \sin\ nx\ dx = \int \sin\ 3x\ dx = -\frac13\ \cos\ 3x + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n \neq 0,\)

\[\int \cos\ nx\ dx = \frac{1}{n}\ \sin\ nx + C.\]

What is the indefinite integral of \(\cos\ 4x\)?

Since \(nx=4x \implies n=4\) in this case, we have \[\int \cos\ nx\ dx = \int \cos\ 4x\ dx = \frac14\ \sin\ 4x + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n, |\sec\ nx| \neq 0,\)

\[\int \tan\ nx\ dx = \frac{1}{n}\ln |\sec\ nx| + C.\]

What is the indefinite integral of \(\tan\ 5x\)?

Since \(nx=5x \implies n=5\) in this case, we have

\[\int \tan\ nx\ dx = \int \tan\ 5x\ dx = \frac15\ln |\sec\ 5x| + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n, |\csc\ nx - \cot\ nx| \neq 0,\)

\[\int \csc\ nx\ dx = \frac{1}{n}\ln |\csc\ nx - \cot\ nx| + C.\]

What is the indefinite integral of \(\csc\ 6x\)?

Since \(nx=6x \implies n=6\) in this case, we have \[\int \csc\ nx\ dx = \int \csc\ 6x\ dx = \frac16\ln |\csc\ 6x - \cot\ 6x| + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n \neq 0,\)

\[\int \csc^2\ nx\ dx = -\frac{1}{n}\cot\ nx + C.\]

What is the indefinite integral of \(\csc^2\ 7x\)?

Since \(nx=7x \implies n=7\) in this case, we have

\[\int \csc^2\ nx\ dx = \int \csc^2\ 7x\ dx = \frac17\cot\ 7x + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n \neq 0,\)

\[\int \csc\ (nx)\cot\ nx\ dx = -\frac{1}{n}\csc\ nx + C.\]

What is the indefinite integral of \(\csc\ (8x)\ \cot\ 8x\)?

Since \(nx=8x \implies n=8\) in this case, we have \[\int \csc\ (nx)\cot\ nx\ dx = \int \csc\ (8x)\ \cot\ 8x\ dx = -\frac18\csc\ 8x + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n, |\sec\ nx + \tan\ nx| \neq 0,\)

\[\int \sec\ nx\ dx = \frac{1}{n}\ln |\sec\ nx + \tan\ nx| + C.\]

What is the indefinite integral of \(\sec\ 9x\)?

Since \(nx=9x \implies n=9\) in this case, we have

\[\int \sec\ nx\ dx = \int \sec\ 9x\ dx = \frac19\ln |\sec\ 9x + \tan\ 9x| + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n \neq 0,\)

\[\int \sec^2\ nx\ dx = \frac{1}{n}\tan\ nx + C.\]

What is the indefinite integral of \(\sec^2\ 10x\)?

Since \(nx=10x \implies n=10\) in this case, we have

\[\int \sec^2\ nx\ dx = \int \sec^2\ 10x\ dx = \frac{1}{10}\tan\ 10x + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n \neq 0,\)

\[\int \sec\ (nx)\ \tan\ nx\ dx = \frac{1}{n}\ \sec\ nx + C.\]

What is the indefinite integral of \(\sec\ 11x\)?

Since \(nx=11x \implies n=11\) in this case, we have

\[\int \sec\ (nx)\ \tan\ nx\ dx = \int \sec\ (11x)\ \tan\ (11x)\ dx = \frac{1}{11}\sec\ 11x + C.\ _\square\]

\[\]

Given an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n, |\sin\ nx| \neq 0,\)

\[\int \cot\ nx\ dx = \frac{1}{n}\ln |\sin\ nx| + C.\]

What is the indefinite integral of \(cot\ 12x\)?

Since \(nx=12x \implies n=12\) in this case

\[\int \cot\ nx\ dx = \int \cot\ 12x\ dx = \frac{1}{12}\ln |\sin\ 12x| + C.\ _\square\]

## Advanced Integration Formulae

The

exponential constant rulestates that, for arbitrary constants \(a\) and \(n\) and an arbitrary variable \(x\) with \(a,n,x \neq 0,\)

\[\int a^{nx}\ dx = \frac{a^{nx}}{n\ ln\ a} + C.\]

What is the indefinite integral of \(13^{4x}\)?

Since \(a=13\) and \(nx=4x \implies n=4\) in this case, we have

\[\int a^{nx}\ dx = \int 13^{4x} dx = \frac{13^{4x}}{4\ ln\ 13} + C.\ _\square\]

\[\]

Given arbitrary constants \(a\) and \(n\) and an arbitrary variable \(x\) with \(a,n,x \neq 0,\) we can derive theintegral of \(e^{nx}\)\(\big(a^{nx}=e^{nx} \implies a=e\big),\) where \(e\) is the base of the natural log, as follows:

\[\int a^{nx}\ dx = \int e^{nx}\ dx = \frac{e^{nx}}{n\ ln\ e} + C = \frac{e^{nx}}n + C.\] Note that \(\ln e = 1.\)

What is the indefinite integral of \(e^{14x}\)?

Since \(nx=14x \implies n=14\) in this case,

\[\int e^{nx}\ dx = \int e^{14x} dx = \frac{e^{14x}}{14} + C.\ _\square\]

\[\]

Thenatural log rulestates that, for an arbitrary constant \(n\) and an arbitrary variable \(x\) with \(n,x \neq 0,\)

\[\int \ln nx\ dx = x\ln |x| - x + C.\]

What is the indefinite integral of \(\ln\ 15x\)?

Since \(nx=15x \implies n=15\) in this case, we have

\[\int \ln nx\ dx = \int \ln 15x\ dx = x\ \ln |x| - x + C.\ _\square\]

However, other integrals, such as \(\int xe^x dx\) and \(\int \sin x \cos x\ dx\), may not be ordinarily evaluated.

For this reason, we need to use appropriate techniques to make the evaluation of such integrals possible.

## U-Substitution

**U-substitution**, a.k.a. the (integration) chain rule, can be done in one of the following 2 methods:

Method 1: Direct Substitution

- Identify which part of the integrand is the original function \(f(x)\) of the other \(\big(\)its derivative \(f'(x)\big),\) and let it be \(u:\) \(u = f(x).\)
- Identify which part is the derivative of \(u\), and let it be \(du: du = f'(x)\ dx.\)
- Replace the integrand in terms of \(u.\)
- Evaluate the resulting integral from (3) in terms of \(u\) and \(du.\)
- Substitute the value of \(u\) in the result from (4) to convert it into an expression in terms of \(x.\)

What is the indefinite integral of \(\big(2x + 4x^2\big)(2 + 8x)?\)

We can write this as \(\int \big(2x + 4x^2\big)(2 + 8x)\, dx.\)

Let \(u=f(x)= 2x + 4x^2,\) then \(du= f'(x)\ dx = (2 + 8x)\ dx,\) which gives \[\int \big(2x + 4x^2\big)(2 + 8x)\ dx = \int u\ du = \frac{u^{1+1}}{1+1} + C = \frac{u^2}2 + C = \frac{(2x + 4x^2)^2}2 + C.\ _\square\]

Method 2: Indirect Substitution

- Identify which part of the integrand is the "more complicated" function \(f(x)\), and let it be \(u:\) \(u = f(x).\)
- Identify which part is the "less complicated" function \(g(x)\), and transform it in terms of \(u.\)
- Replace the integrand in terms of \(u.\)
- Evaluate the resulting integral from (3) in terms of \(u\) and \(du.\)
- Substitute the value of \(u\) in the result from (4) to convert it into an expression in terms of \(x.\)

What is the indefinite integral of \(\big(3x^3\big)(x+1)^8\)?

We can write this as \(\int \big(3x^3\big)(x+1)^8\, dx\).

Let \(u=f(x)= x+1,\) then \(du= 1\ dx \implies du=dx.\)

Also let \(g(x)=3x^3,\) then \(u= x+1 \implies x= u-1 \implies 3x^3= 3(u-1)^3,\) which gives\[\begin{align} \int 3(u-1)^3\ u^8\ dx &= 3 \int \big(u^3 - 3u^2 + 3u - 1\big)\ u^8\ du \\ &= 3 \int \big(u^{11} - 3u^{10} + 3u^9 - u^8\big)\ du \\ &= 3\ \left(\int u^{11}\ du - \int 3u^{10}\ du + \int 3u^9\ du - \int u^8\ du\right) \\ &= 3\ \left(\frac{u^{11+1}}{11+1} - 3\frac{u^{10+1}}{10+1} + 3\frac{u^{9+1}}{9+1} - \frac{u^{8+1}}{8+1} + C\right) \\ &= 3\ \left(\frac{u^{11+1}}{11+1} - \frac3{10+1}u^{10+1} + \frac3{9+1}u^{9+1} - \frac{u^{8+1}}{8+1} + C\right) \\ &= 3\ \left(\frac{u^{12}}{12} - \frac3{11}u^{11} + \frac3{10}u^{10} - \frac{u^9}9 + C\right) \\ &= \frac3{12}u^{12} - \frac{3^2}{11}u^{11} + \frac{3^2}{10}u^{10} - \frac39u^9 + C \\\\ &= \frac14u^{12} - \frac9{11}u^{11} + \frac9{10}u^{10} - \frac13u^9 + C \\\\ &= \frac{u^{12}}4 - \frac9{11}u^{11} + \frac9{10}u^{10} - \frac{u^9}3 + C \\\\ &= \frac{(x+1)^{12}}4 - \frac9{11}(x+1)^{11} + \frac9{10}(x+1)^{10} - \frac{(x+1)^9}3 + C.\ _\square \end{align}\]

## Integration by Parts

Another technique that we can use to evaluate tricky-looking integrals is

integration by parts. This is similar to the \(u\)-substitution methods. Given arbitrary functions \(f(x)\) and \(g(x)\) with \(u=f(x)\) and \(v' = g(x)\ dx\), and an arbitrary differential \(dx,\)

\[\int uv'\ dx = uv - \int u'v\ dx.\] This formula was derived from theproduct rulein differentiation.

- Identify which part of the integrand is \(u\): this may be the "written" part \(f(x)\). However, if there are 2 arbitrary functions \(f(x)\) and \(g(x)\) multiplied together, let the function that is easier to differentiate be \(u.\)
- Identify which part of the integrand is \(v'\): this may be the "non-written" part \(1\). However, if there are 2 arbitrary functions \(f(x)\) and \(g(x)\) multiplied together, let the function that is easier to integrate be \(v'.\)
- Integrate the value of \(v'\) to get \(v.\)
- Integrate \(u'v\) for the subtrahend \(\int u'v.\)
- Subtract the result in (4) from the value of \(uv.\)
- Substitute the values of \(u\), \(v\) and \(u'\) in the result from (5).

What is the indefinite integral of \(x^2\ln x\) \((\)with \(x=|n|,\) where \(n \neq 0)?\)

We can write this as \(\int x^2\ln x\ dx.\)

Let \(u= \ln x,\) then \(u'=x^{-1}.\)

Also let \(v'=x^2,\) then \(\frac{dv}{dx}=x^2, \int \frac{dv}{dx}\ dx = \int dv = v= \int x^2\ dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}3 + C \implies v=\frac{x^3}3,\) which implies\[\begin{align} \int x^2\ln x\ dx &= \int uv'\ dx \\ &= uv - \int u'v\ dx \\ &= \ln x\ \frac{x^3}3 - \int \frac{1}{x} \cdot \frac{x^3}3\ dx \\ &= \frac{x^3}3 \ln\ x - \int \frac{x^{3-1}}3\ dx \\ &= \frac{x^3}3 \ln\ x - \int \frac{x^2}3\ dx \\ &= \frac{x^3}3 \ln\ x - \frac13 \int x^2\ dx \\ &= \frac{x^3}3 \ln\ x - \frac13 \cdot \frac{x^{2+1}}{2+1} + C \\ &= \frac{x^3}3 \ln\ x - \frac13 \cdot \frac{x^3}3 + C \\ &= \frac{x^3}3 \ln\ x - \frac{x^3}{9} + C.\ _\square \end{align}\]

**Cite as:**Integration Techniques.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/integration-techniques/