Integral formulas and rules reference

Integral Techniques

u Substitution

\[\int^b_a f[g(x)g^\prime(x)] \qquad u = g(x);\,\,du = g^\prime(x)\]

\[\equiv \int^{g(b)}_{g(a)} = f(u)du\]

When you do a u sub you have to either change the limits of integration to be consistent with the transformation or before you calculate the numbers with the limits convert all the "u's" back to g(x)

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Integration by Parts

\[\int udv = uv-\int vdu\]

Use \(\text{LIPET}\) to calculate u and v
- L = Logarithms
- I = Inverse trigonometry
- P = Polynomial
- E = Exponential
- T = Trigonometry

While not always guaranteed to work its a decent rule of thumb and a good starting place
You will ocasionally have to do multiple IBP's to solve an integral, as well as canceling parts when it goes into recursion. To speed this process you can use the tabular method

\[ \begin{array}{|ccc|} \textcolor{red}{u} && dv \\ & \textcolor{red}{+\searrow} &\\ \textcolor{yellow}{u^{\prime}} && \textcolor{red}{\int dv} \\ &\textcolor{yellow}{-\searrow}\\ \textcolor{magenta}{{u^{\prime\prime}}} && \textcolor{yellow}{\iint dv}\\ &\textcolor{magenta}{+\searrow}\\\textcolor{lime} {u^{\prime\prime\prime}}&& \textcolor{magenta}{\iiint dv} \\ &\textcolor{lime}{-\searrow}\\\vdots &&\textcolor{lime}{\vdots} \\\end{array}\]

By adding the multiplied diagonals and alternating signs, this is equivalent to doing an integration by parts, and can speed up the process by quite a bit

\[\int udv =\left[u\int dv\right] - \left[u^{\prime}\iint dv\right] + \left[u^{\prime\prime}\iiint dv\right] - \cdots \]

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Trigonometric Substitution

Form Looks Like Substitution

\(\large\sqrt {{b^2}{x^2} - {a^2}}\) \(\large{\sec ^2}\theta - 1 = \tan^{2} \theta\) \(\large x = \frac{a}{b}\sec \theta\)

\(\large\sqrt {{a^2} - {b^2}{x^2}}\) \(\large 1 - {\sin ^2}\theta = {\cos ^2}\theta\) \(\large x = \frac{a}{b}\sin \theta\)

\(\large \sqrt {{a^2} + {b^2}{x^2}}\) \(\large {\tan ^2}\theta + 1 = {\sec ^2}\theta\) \(\large x = \frac{a}{b}\tan \theta\)

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Trigonometric Integrals

\[\int \sin^{m}(x)cos^{n}(x)\,dx\]

if Strategy

\(m\) odd and positive, \(n\) real Split off \(\sin(x)\), rewrite the resulting even power of \(\sin(x)\) in terms of \(\cos(x)\), use \(u = \cos(x)\)

\(n\) odd and positive, \(m\) real Split off \(\cos(x)\), rewrite the resulting even power of \(\cos(x)\) in terms of \(\sin(x)\), use \(u = \sin(x)\)

\(m\) and \(n\) both even, nonnegative integers Use half-angle formulas to transform the integrand into a polynomial in \(\cos(2x)\) and apply the preceding strategies once again to powers of \(\cos(2x)\) greater than 1.

\[\int\tan^{m}(x)\sec^{n}(x)\,dx\]

if Strategy

\(m\) odd and positive, \(n\) real Split off \(\sec(x)\tan(x)\), rewrite the remaining even power of \(\tan(x)\) in terms of \(\sec(x)\), use \(u = \sec(x)\)

\(n\) even and positive, \(m\) real Split off \(\sec^{2}(x)\), rewrite the remaining even power of \(\sec(x)\) in terms of \(\tan(x)\), use \(u = \tan(x)\)

\(m\) and \(n\) both even, nonnegative integers Rewrite the even power of \(\tan(x)\) in terms of \(\sec(x)\) to produce a polynomial in \(\sec(x)\); apply the 4th reduction formula below to each term

Reduction formulas
(note: can only be used if function is by itself)

\[\int \sin ^n x d x=-\frac{\sin ^{n-1} x \cos x}{n}+\frac{n-1}{n} \int \sin ^{n-2} x d x\]

\[\int \cos ^n x d x=\frac{\cos ^{n-1} x \sin x}{n}+\frac{n-1}{n} \int \cos ^{n-2} x d x\]

\[\int \tan ^n x d x=\frac{\tan ^{n-1} x}{n-1}-\int \tan ^{n-2} x d x, n \neq 1\]

\[\int \sec ^n x d x=\frac{\sec ^{n-2} x \tan x}{n-1}+\frac{n-2}{n-1} \int \sec ^{n-2} x d x, n \neq 1\]

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Partial Fractions

Factor in denominator Term in partial fraction decomposition

\(\large ax + b\) \(\large\displaystyle \frac{A}{{ax + b}}\)

\(\large{\left( {ax + b} \right)^k}\) \(\large\displaystyle \frac{{{A_1}}}{{ax + b}} + \frac{{{A_2}}}{{{{\left( {ax + b} \right)}^2}}} + \cdots + \frac{{{A_k}}}{{{{\left( {ax + b} \right)}^k}}}\)

\(\large a{x^2} + bx + c\) \(\large \displaystyle \frac{{Ax + B}}{{a{x^2} + bx + c}}\)

\(\large {\left( {a{x^2} + bx + c} \right)^k}\) \(\large\displaystyle \frac{{{A_1}x + {B_1}}}{{a{x^2} + bx + c}} + \frac{{{A_2}x + {B_2}}}{{{{\left( {a{x^2} + bx + c} \right)}^2}}} + \cdots + \frac{{{A_k}x + {B_k}}}{{{{\left( {a{x^2} + bx + c} \right)}^k}}}\)

Polynomials

\[\int dx = x +c \qquad \int kdx = kx+c \qquad\int x^ndx = \frac{x^{n+1}}{n+1} +c\]

\[\int \frac{1}{x}dx = \ln|x|+c \qquad \int \frac{1}{ax+b}dx = \frac{ln|ax+b|}{a} +c\]

Trigonometry

\[\begin{align*}\int \sin u \,du=-\cos u +c && \int \csc u \, du =\ln|\csc u-\cot u|+c\end{align*}\]

\[\begin{align*}\int \cos u \,du = \sin u + c && \int \sec u\,du = \ln|\sec u + \tan u|+c\end{align*}\]

\[\begin{align*}\int\tan u \, du = \ln|\sec u| +c && \int\cot u \, du = \ln |\sin u| + c \end{align*}\]

\[\begin{align*}\int\sec u \tan u \, du = \sec u + c && \int \csc u \cot u \, du = -\csc u +c\end{align*}\]

\[\begin{align*}\int \sec^2u\, du = \tan u +c && \int\csc^2u\,du = -\cot u+c \end{align*}\]

\[\int \sec^3u\,du = \frac{1}{2}(\sec u \tan u + \ln|\sec u+\tan u|)+c\]

\[\int \csc^3u\,du = \frac{1}{2}(-\csc u\cot u+\ln|\csc u-\cot u|)+c\]

Exponential/Logarithmic

\[\begin{align*}\int e^u \, du = e^u + c && \int a^u\,du = \frac{a^u}{\ln a}+c \end{align*}\]

\[\begin{align*}\int \ln u \, du = u\ln(u)-u+c && \int \frac{1}{u\ln u}\,du = \ln|\ln u| +c \end{align*}\]

\[\begin{align*} \int ue^u\,du = (u-1)e^u+c \end{align*}\]

\[\int e^{au}\sin(bu)\,du = \frac{e^{au}}{a^2+b^2}(a\sin(bu)-b\cos(bu))+c\]

\[\begin{align*}\int e^{au}\cos(bu)\,du = \frac{e^{au}}{a^2+b^2}(a\cos(bu)+b\sin(bu))+c && \end{align*}\]

Inverse Trigonometry

\[\int \sin^{-1}u\,du = u\sin^{-1}u+\sqrt{1-u^2}+c\]

\[\int \frac{1}{\sqrt{a^2-u^2}}du = \sin ^{-1}(\frac{u}{a})+c\]

\[\int \cos^{-1}u\, du = u \cos^{-1}u-\sqrt{1-u^2}+c\]

\[\int\frac{1}{u\sqrt{u^2-a^2}}\,du = \frac{1}{a}\sec^{-1}(\frac{u}{a}) +c\]

\[\int \tan^{-1}u\,du = u\tan^{-1}u-\frac{1}{2}\ln(1+u^2)+c\]

\[\int\frac{1}{a^2+u^2}\,du = \frac{1}{a}\tan^{-1}(\frac{u}{a})+c \]

Hyperbolic Trigonometry

\[\begin{align*} \int\sinh u\,du = \cosh u+c && \int\operatorname{csch}^2 u \,du = -\coth u+c\end{align*}\]

\[\begin{align*} \int\cosh u\,du=\sinh u +c && \int\operatorname{sech}^2u\,du = \tanh u + c \end{align*}\]

\[\begin{align*} \int \tanh u\,du = \ln(\cosh u)+c && \int \coth u\,du = \ln(\cosh x)+c\end{align*}\]

\[\begin{align*}\int\operatorname{sech}u\,du = \tan^{-1}|\sinh u|+c && \int\operatorname{csch} u\,du = \ln|\tanh(\frac{u}{2})|+c\end{align*}\]

\[\begin{align*} \int \operatorname{csch}u \coth u \,du = -\operatorname{csch}u+c && \operatorname{sech}u \tanh u\,du = -\operatorname{sech}u+c \end{align*}\]

Form	Looks Like	Substitution
\(\large\sqrt {{b^2}{x^2} - {a^2}}\)	\(\large{\sec ^2}\theta - 1 = \tan^{2} \theta\)	\(\large x = \frac{a}{b}\sec \theta\)
\(\large\sqrt {{a^2} - {b^2}{x^2}}\)	\(\large 1 - {\sin ^2}\theta = {\cos ^2}\theta\)	\(\large x = \frac{a}{b}\sin \theta\)
\(\large \sqrt {{a^2} + {b^2}{x^2}}\)	\(\large {\tan ^2}\theta + 1 = {\sec ^2}\theta\)	\(\large x = \frac{a}{b}\tan \theta\)

if	Strategy
\(m\) odd and positive, \(n\) real	Split off \(\sin(x)\), rewrite the resulting even power of \(\sin(x)\) in terms of \(\cos(x)\), use \(u = \cos(x)\)
\(n\) odd and positive, \(m\) real	Split off \(\cos(x)\), rewrite the resulting even power of \(\cos(x)\) in terms of \(\sin(x)\), use \(u = \sin(x)\)
\(m\) and \(n\) both even, nonnegative integers	Use half-angle formulas to transform the integrand into a polynomial in \(\cos(2x)\) and apply the preceding strategies once again to powers of \(\cos(2x)\) greater than 1.

if	Strategy
\(m\) odd and positive, \(n\) real	Split off \(\sec(x)\tan(x)\), rewrite the remaining even power of \(\tan(x)\) in terms of \(\sec(x)\), use \(u = \sec(x)\)
\(n\) even and positive, \(m\) real	Split off \(\sec^{2}(x)\), rewrite the remaining even power of \(\sec(x)\) in terms of \(\tan(x)\), use \(u = \tan(x)\)
\(m\) and \(n\) both even, nonnegative integers	Rewrite the even power of \(\tan(x)\) in terms of \(\sec(x)\) to produce a polynomial in \(\sec(x)\); apply the 4th reduction formula below to each term

Factor in denominator	Term in partial fraction decomposition
\(\large ax + b\)	\(\large\displaystyle \frac{A}{{ax + b}}\)
\(\large{\left( {ax + b} \right)^k}\)	\(\large\displaystyle \frac{{{A_1}}}{{ax + b}} + \frac{{{A_2}}}{{{{\left( {ax + b} \right)}^2}}} + \cdots + \frac{{{A_k}}}{{{{\left( {ax + b} \right)}^k}}}\)
\(\large a{x^2} + bx + c\)	\(\large \displaystyle \frac{{Ax + B}}{{a{x^2} + bx + c}}\)
\(\large {\left( {a{x^2} + bx + c} \right)^k}\)	\(\large\displaystyle \frac{{{A_1}x + {B_1}}}{{a{x^2} + bx + c}} + \frac{{{A_2}x + {B_2}}}{{{{\left( {a{x^2} + bx + c} \right)}^2}}} + \cdots + \frac{{{A_k}x + {B_k}}}{{{{\left( {a{x^2} + bx + c} \right)}^k}}}\)