Integration Techniques

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

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

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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 \(666x\)?

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

Here are fundamental theorems for simple evaluation of integrals.

The constant rule states that for an arbitrary constant \(a\)
\[\int a\ dx = ax + C.\]

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

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Since \(a=1\) in this case,
\[\int a\ dx = \int 1\ dx = \int dx = 1x + C = x + C.\ _\square\]

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The constant function rule, more formally known as the constant 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 \(2x\)?

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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.

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The sum and difference rules state 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 \(2x + 3x^2\)?

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

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The power rule states 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 \(x\)?

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Since \(x^n=x=x^1 \implies n=1\) in this case, we have
\[\int x^n\ dx = \int x\ dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C.\ _\square\]

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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 \(\frac1{2x}\)?

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

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 \(\sin\ 3x\)?

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

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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 \(\cos\ 4x\)?

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

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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 \(\tan\ 5x\)?

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Since \(nx=5x \implies n=5\) in this case, we have
\[\int \tan\ nx\ dx = \int \tan\ 5x\ dx = \frac15\ln |\sec\ 5x| + C.\ _\square\]

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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 \(\csc\ 6x\)?

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

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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 \(\csc^2\ 7x\)?

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Since \(nx=7x \implies n=7\) in this case, we have
\[\int \csc^2\ nx\ dx = \int \csc^2\ 7x\ dx = \frac17\cot\ 7x + C.\ _\square\]

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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 \(\csc\ (8x)\ \cot\ 8x\)?

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

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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 \(\sec\ 9x\)?

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Since \(nx=9x \implies n=9\) in this case, we have
\[\int \sec\ nx\ dx = \int \sec\ 9x\ dx = \frac19\ln |\sec\ 9x + \tan\ 9x| + C.\ _\square\]

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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 \(\sec^2\ 10x\)?

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Since \(nx=10x \implies n=10\) in this case, we have
\[\int \sec^2\ nx\ dx = \int \sec^2\ 10x\ dx = \frac{1}{10}\tan\ 10x + C.\ _\square\]

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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 \(\sec\ 11x\)?

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Since \(nx=11x \implies n=11\) in this case, we have
\[\int \sec\ (nx)\ \tan\ nx\ dx = \int \sec\ (11x)\ \tan\ (11x)\ dx = \frac{1}{11}\sec\ 11x + C.\ _\square\]

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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 \(cot\ 12x\)?

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Since \(nx=12x \implies n=12\) in this case
\[\int \cot\ nx\ dx = \int \cot\ 12x\ dx = \frac{1}{12}\ln |\sin\ 12x| + C.\ _\square\]

The exponential constant rule states 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 \(13^{4x}\)?

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Since \(a=13\) and \(nx=4x \implies n=4\) in this case, we have
\[\int a^{nx}\ dx = \int 13^{4x} dx = \frac{13^{4x}}{4\ ln\ 13} + C.\ _\square\]

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Given arbitrary constants \(a\) and \(n\) and an arbitrary variable \(x\) with \(a,n,x \neq 0,\) we can derive the integral 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 \(e^{14x}\)?

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Since \(nx=14x \implies n=14\) in this case,
\[\int e^{nx}\ dx = \int e^{14x} dx = \frac{e^{14x}}{14} + C.\ _\square\]

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The natural log rule states 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 \(\ln\ 15x\)?

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

Method 1: Direct Substitution

1. 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 \(u:\) \(u = f(x).\)
2. Identify which part is the derivative of \(u\), and let it be \(du: du = f'(x)\ dx.\)
3. Replace the integrand in terms of \(u.\)
4. Evaluate the resulting integral from (3) in terms of \(u\) and \(du.\)
5. 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 \(\big(2x + 4x^2\big)(2 + 8x)?\)

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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

1. Identify which part of the integrand is the "more complicated" function \(f(x)\), and let it be \(u:\) \(u = f(x).\)
2. Identify which part is the "less complicated" function \(g(x)\), and transform it in terms of \(u.\)
3. Replace the integrand in terms of \(u.\)
4. Evaluate the resulting integral from (3) in terms of \(u\) and \(du.\)
5. 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 \(\big(3x^3\big)(x+1)^8\)?

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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

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

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 the product rule in differentiation.

1. 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 \(u.\)
2. 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 \(v'.\)
3. Integrate the value of \(v'\) to get \(v.\)
4. Integrate \(u'v\) for the subtrahend \(\int u'v.\)
5. Subtract the result in (4) from the value of \(uv.\)
6. 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 \(n \neq 0)?\)

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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

\[\begin{align} \int x^2\ln x\ dx &= \int uv'\ dx \\ &= uv - \int u'v\ dx \\ &= \ln x\ \frac{x^3}3 - \int \frac{1}{x} \cdot \frac{x^3}3\ dx \\ &= \frac{x^3}3 \ln\ x - \int \frac{x^{3-1}}3\ dx \\ &= \frac{x^3}3 \ln\ x - \int \frac{x^2}3\ dx \\ &= \frac{x^3}3 \ln\ x - \frac13 \int x^2\ dx \\ &= \frac{x^3}3 \ln\ x - \frac13 \cdot \frac{x^{2+1}}{2+1} + C \\ &= \frac{x^3}3 \ln\ x - \frac13 \cdot \frac{x^3}3 + C \\ &= \frac{x^3}3 \ln\ x - \frac{x^3}{9} + C.\ _\square \end{align}\]