Definitions [1]
Integration by substitution is a method in which we replace a part of the integral by a new variable to simplify the integration.
Formulae [4]
| 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 \[-cosec^{-1}x\] |
| \[\frac{1}{x^2+a^2}\] | \[\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)\] |
| \[\frac{1}{x^2-a^2}\] | \[\frac{1}{2\mathrm{a}}\mathrm{log}\left|\frac{x-\mathrm{a}}{x+\mathrm{a}}\right|\] |
| \[\frac{1}{\mathbf{a}^2-x^2}\] | \[\frac{1}{2\mathrm{a}}\mathrm{log}\left|\frac{\mathrm{a}+x}{\mathrm{a}-x}\right|\] |
| \[\frac{1}{\sqrt{x^2+a^2}}\] | \[\log\left|x+\sqrt{x^{2}+\mathrm{a}^{2}}\right|\] |
| \[\frac{1}{\sqrt{x^2-a^2}}\] | \[\log\left|x+\sqrt{x^{2}-a^{2}}\right|\] |
| \[\frac{1}{\sqrt{a^2-x^2}}\] | \[\sin^{-1}\left(\frac{x}{a}\right)\] |
| \[\frac{1}{x\sqrt{x^2-a^2}}\] | \[\frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right)\] |
| \[\sqrt{a^{2}-x^{2}}\] | \[\frac{x}{2}\sqrt{\mathrm{a}^{2}-x^{2}}+\frac{\mathrm{a}^{2}}{2}\sin^{-1}\left(\frac{x}{\mathrm{a}}\right)\] |
| \[\sqrt{x^2+a^2}\] | \[\frac{x}{2}\sqrt{x^{2}+\mathrm{a}^{2}}+\frac{\mathrm{a}^{2}}{2}\log\left|x+\sqrt{x^{2}+\mathrm{a}^{2}}\right|\] |
| \[\sqrt{x^{2}-a^{2}}\] | \[\frac{x}{2}\sqrt{x^{2}-\mathrm{a}^{2}}-\frac{\mathrm{a}^{2}}{2}\log\left|x+\sqrt{x^{2}-\mathrm{a}^{2}}\right|\] |
\[\int\left(\mathrm{u.v}\right)\mathrm{dx}=\mathrm{u}\int\mathrm{v}\mathrm{dx}-\int\left(\frac{\mathrm{du}}{\mathrm{dx}}\right).\left(\int\mathrm{v}\mathrm{dx}\right)\mathrm{dx}\]
Special Result:
∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C
(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
| Sr. No. | Integrand Form | Substitution |
|---|---|---|
| i | \[\sqrt{\mathrm{a}^2-x^2},\frac{1}{\sqrt{\mathrm{a}^2-x^2}},\mathrm{a}^2-x^2\] | x = a sinθ or x = a cosθ |
| ii | \[\sqrt{x^2+\mathrm{a}^2},\frac{1}{\sqrt{x^2+\mathrm{a}^2}},x^2+\mathrm{a}^2\] | x = a tanθ |
| iii | \[\sqrt{x^{2}-a^{2}},\frac{1}{\sqrt{x^{2}-a^{2}}},x^{2}-a^{2}\] | x = a secθ |
| iv | \[\sqrt{\frac{x}{a+x}},\sqrt{\frac{a+x}{x}},\]\[\sqrt{x(a+x)},\frac{1}{\sqrt{x(a+x)}}\] | x = a tan²θ |
| v | \[\sqrt{\frac{x}{a-x}},\sqrt{\frac{a-x}{x}},\]\[\sqrt{x(a-x)},\frac{1}{\sqrt{x(a-x)}}\] | x = a sin²θ |
| vi | \[\sqrt{\frac{x}{x-a}},\sqrt{\frac{x-a}{x}},\]\[\sqrt{x(x-\mathrm{a})},\frac{1}{\sqrt{x(x-\mathrm{a})}}\] | x = a sec²θ |
| vii | \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}},\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\] | x = a cos 2θ |
| viii | \[\sqrt{\frac{x-\alpha}{\beta-x}},\sqrt{(x-\alpha)(\beta-x)},\]\[(\beta>\alpha)\] | x = α cos²θ + β sin²θ |
First function should be chosen in the following order of preference:
L → Logarithmic function
I → Inverse trigonometric function
A → Algebraic function
T → Trigonometric function
E → Exponential function
Note:
For the integration of logarithmic or inverse trigonometric functions alone, take unity (1) as the second function.
Standard forms:
i) \[\int\sqrt{x^{2}+a^{2}}dx=\frac{1}{2}\left[ \begin{array} {c}{x\sqrt{x^{2}+a^{2}}} {+a^{2}\log|x+\sqrt{x^{2}+a^{2}|}} \end{array}\right]+C\]
ii) \[\int\sqrt{a^{2}-x^{2}}dx=\frac{1}{2}\left[x\sqrt{a^{2}-x^{2}}+a^{2}\sin^{-1}\left(\frac{x}{a}\right)\right]+C\]
iii) \[\int\sqrt{x^{2}-a^{2}}dx=\frac{1}{2}[x\sqrt{x^{2}-a^{2}}-a^{2}\log|x+\sqrt{x^{2}-a^{2}}|]\] + C
| Type | Rational Form | Partial Form |
|---|---|---|
| Type I (Non-repeated linear factors) | \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\] | \[\frac{\mathrm{A}}{x-\mathrm{a}}+\frac{\mathrm{B}}{x-\mathrm{b}}\] |
| \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x-\mathrm{b})(x-\mathrm{c})}\] | \[\frac{\mathrm{A}}{x-\mathrm{a}}+\frac{\mathrm{B}}{x-\mathrm{b}}+\frac{\mathrm{C}}{x-\mathrm{c}}\] | |
| Type II (Repeated linear factors) | \[\frac{\mathrm{p}x+\mathrm{q}}{\left(x-\mathrm{a}\right)^2}\] | \[\frac{\mathrm{A}}{(x-\mathrm{a})}+\frac{\mathrm{B}}{(x-\mathrm{a})^{2}}\] |
| \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{\left(x-\mathrm{a}\right)^{2}\left(x-\mathrm{b}\right)}\] | \[\frac{\mathrm{A}}{(x-\mathrm{a})}+\frac{\mathrm{B}}{(x-\mathrm{a})^{2}}+\frac{\mathrm{C}}{(x-\mathrm{b})}\] | |
| Type III (Linear × Quadratic) | \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\] | \[\frac{\mathrm{A}}{(x-\mathrm{a})}+\frac{\mathrm{B}x+\mathrm{C}}{(x^{2}+\mathrm{b}x+\mathrm{c})}\] |
