Definitions [3]
Integration by substitution is a method in which we replace a part of the integral by a new variable to simplify the integration.
General Formula:
If \[x = g(t), \ dx = g'(t) dt\] then \[\int f(x) dx = \int f(g(t))g'(t) dt\]
If two functions are written in the form uu and dvdv, then integration by parts is based on the product rule of differentiation.
\[\int\left(\mathrm{u.v}\right)\mathrm{dx}=\mathrm{u}\int\mathrm{v}\mathrm{dx}-\int\left(\frac{\mathrm{du}}{\mathrm{dx}}\right).\left(\int\mathrm{v}\mathrm{dx}\right)\mathrm{dx}\]
Integration by partial fractions is a method used to integrate rational functions, that is, functions of the form
Formulae [3]
| f(x) | ∫ f(x) dx |
|---|---|
| xⁿ | \[\frac{x^{\mathrm{n+1}}}{\mathrm{n+1}}\], n ≠ -1 |
| \[\frac{1}{x}\] | log |
| eˣ | eˣ |
| aˣ | \[\frac{a^x}{\log a}\] (a ≠ 1, a > 0) |
| log x | x(log x − 1) |
| sin x | −cos x |
| cos x | sin x |
| sec² x | tan x |
| cosec² x | −cot x |
| sec x tan x | sec x |
| cosec x cot x | −cosec x |
| tan x | \[-\log|\cos x|\] or \[\log\left|\sec x\right|\] |
| cot x | \[\log|\sin x|\] or \[-\log|\operatorname{cosec}x|\] |
| sec x | \[\log|\sec x+\tan x|\] or \[\log\left|\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\right|\] |
| cosec x | \[\log|\operatorname{cosec}x-\cot x|\] or \[\log\left|\tan\frac{x}{2}\right|\] |
| \[\frac{1}{\sqrt{1-x^2}}\] | sin⁻¹ x or cos⁻¹ x |
| \[\frac{1}{1+x^2}\] | tan⁻¹ x or −cot⁻¹ x |
| \[\frac{1}{|x|\sqrt{x^2-1}}\] | \[sec^{-1}x\] or \[-cosec^{-1}x\] |
| \[\frac{1}{x^2+a^2}\] | \[\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)\] |
| \[\frac{1}{x^2-a^2}\] | \[\frac{1}{2\mathrm{a}}\mathrm{log}\left|\frac{x-\mathrm{a}}{x+\mathrm{a}}\right|\] |
| \[\frac{1}{\mathbf{a}^2-x^2}\] | \[\frac{1}{2\mathrm{a}}\mathrm{log}\left|\frac{\mathrm{a}+x}{\mathrm{a}-x}\right|\] |
| \[\frac{1}{\sqrt{x^2+a^2}}\] | \[\log\left|x+\sqrt{x^{2}+\mathrm{a}^{2}}\right|\] |
| \[\frac{1}{\sqrt{x^2-a^2}}\] | \[\log\left|x+\sqrt{x^{2}-a^{2}}\right|\] |
| \[\frac{1}{\sqrt{a^2-x^2}}\] | \[\sin^{-1}\left(\frac{x}{a}\right)\] |
| \[\frac{1}{x\sqrt{x^2-a^2}}\] | \[\frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right)\] |
| \[\sqrt{a^{2}-x^{2}}\] | \[\frac{x}{2}\sqrt{\mathrm{a}^{2}-x^{2}}+\frac{\mathrm{a}^{2}}{2}\sin^{-1}\left(\frac{x}{\mathrm{a}}\right)\] |
| \[\sqrt{x^2+a^2}\] | \[\frac{x}{2}\sqrt{x^{2}+\mathrm{a}^{2}}+\frac{\mathrm{a}^{2}}{2}\log\left|x+\sqrt{x^{2}+\mathrm{a}^{2}}\right|\] |
| \[\sqrt{x^{2}-a^{2}}\] | \[\frac{x}{2}\sqrt{x^{2}-\mathrm{a}^{2}}-\frac{\mathrm{a}^{2}}{2}\log\left|x+\sqrt{x^{2}-\mathrm{a}^{2}}\right|\] |
(i) ∫ tan x dx = − log |cos x| + C = log |sec x| + C
(ii) ∫ cot x dx = log |sin x| + C
(iii) ∫ sec x dx = log |sec x + tan x| + C
(iv) ∫ cosec x dx = log |cosec x − cot x| + C
(i) \[\int\frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C\]
(ii) \[\int\frac{dx}{a^{2}-x^{2}}=\frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C\]
(iii) \[\int\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}+C\]
(iv) \[\int\frac{dx}{\sqrt{x^{2}-a^{2}}}=\log|x+\sqrt{x^{2}-a^{2}}|+C\]
(v) \[\int\frac{dx}{\sqrt{a^{2}-x^{2}}}=\sin^{-1}\frac{x}{a}+C\]
(vi) \[\int\frac{dx}{\sqrt{x^{2}+a^{2}}}=\log|x+\sqrt{x^{2}+a^{2}}|+C\]
Theorems and Laws [1]
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Let I = `int sqrt(a^2 - x^2) dx`
= `int sqrt(a^2 - x^2)*1 dx`
= `sqrt(a^2 - x^2)* int 1 dx - int [d/dx (sqrt(a^2 - x^2))* int 1 dx]dx`
= `sqrt(a^2 - x^2)*x - int [1/(2sqrt(a^2 - x^2))*d/dx (a^2 - x^2)*x]dx`
= `sqrt(a^2 - x^2)*x - int 1/(2sqrt(a^2 - x^2))(0 - 2x)*x dx`
= `sqrt(a^2 - x^2)*x - int (-x)/sqrt(a^2 - x^2)*x dx`
= `xsqrt(a^2 - x^2) - int (a^2 - x^2 - a^2)/sqrt(a^2 - x^2)dx`
= `xsqrt(a^2 - x^2) - int sqrt(a^2 - x^2)dx + a^2 int dx/sqrt(a^2 - x^2)`
= `xsqrt(a^2 - x^2) - I + a^2sin^-1(x/a) + c_1`
∴ 2I = `xsqrt(a^2 - x^2) + a^2sin^-1(x/a) + c_1`
∴ I = `x/2 sqrt(a^2 - x^2) + a^2/2 sin^-1(x/a) + c_1/2`
∴ `int sqrt(a^2 - x^2)dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a) + c`, where `c = c_1/2`.
Key Points
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Integration by substitution is the reverse process of the chain rule.
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Choose the substitution so that the integral becomes simpler, not more complicated.
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Always rewrite both the function and \(dx\) in terms of the new variable.
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For indefinite integrals, back-substitute and add \(C\).
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For definite integrals, limits should also be changed if the solution is continued in the new variable.
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Trigonometric substitution is mainly used for radicals involving \(a^2-x^2\), \(x^2+a^2\), and \(x^2-a^2\).
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Formula:
\[\int u dv = uv - \int v du\] -
Choose u by LIATE
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For log x and inverse trig, multiply by 1
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Repeated parts may be needed for \[e^x \sin x\], \[e^x \cos x\].
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First check whether the rational function is proper or improper.
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Use long division before decomposition if the fraction is improper.
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Factorise the denominator completely before choosing partial fractions.
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For each distinct linear factor, use a constant numerator such as A, B, or C.
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For a repeated linear factor, include every power separately.
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For an irreducible quadratic factor, use a linear numerator of the form Bx + C.
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After decomposition, integrate each term separately using standard formulas.
