Definitions [1]
Definition: Definite integral
A definite integral represents the value of a function accumulated between two limits.
It can also be interpreted geometrically as the net area between the graph of y = f(x) and the x-axis from x = a to x = b.
Formulae [1]
Formula: Integration of Some Standard Functions
| 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 \[-cosec^{-1}x\] |
| \[\frac{1}{x^2+a^2}\] | \[\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)\] |
| \[\frac{1}{x^2-a^2}\] | \[\frac{1}{2\mathrm{a}}\mathrm{log}\left|\frac{x-\mathrm{a}}{x+\mathrm{a}}\right|\] |
| \[\frac{1}{\mathbf{a}^2-x^2}\] | \[\frac{1}{2\mathrm{a}}\mathrm{log}\left|\frac{\mathrm{a}+x}{\mathrm{a}-x}\right|\] |
| \[\frac{1}{\sqrt{x^2+a^2}}\] | \[\log\left|x+\sqrt{x^{2}+\mathrm{a}^{2}}\right|\] |
| \[\frac{1}{\sqrt{x^2-a^2}}\] | \[\log\left|x+\sqrt{x^{2}-a^{2}}\right|\] |
| \[\frac{1}{\sqrt{a^2-x^2}}\] | \[\sin^{-1}\left(\frac{x}{a}\right)\] |
| \[\frac{1}{x\sqrt{x^2-a^2}}\] | \[\frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right)\] |
| \[\sqrt{a^{2}-x^{2}}\] | \[\frac{x}{2}\sqrt{\mathrm{a}^{2}-x^{2}}+\frac{\mathrm{a}^{2}}{2}\sin^{-1}\left(\frac{x}{\mathrm{a}}\right)\] |
| \[\sqrt{x^2+a^2}\] | \[\frac{x}{2}\sqrt{x^{2}+\mathrm{a}^{2}}+\frac{\mathrm{a}^{2}}{2}\log\left|x+\sqrt{x^{2}+\mathrm{a}^{2}}\right|\] |
| \[\sqrt{x^{2}-a^{2}}\] | \[\frac{x}{2}\sqrt{x^{2}-\mathrm{a}^{2}}-\frac{\mathrm{a}^{2}}{2}\log\left|x+\sqrt{x^{2}-\mathrm{a}^{2}}\right|\] |
Key Points
Key Points: Definite Integrals
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Used to find exact accumulated value over a fixed interval.
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Written as \[\int_{a}^{b} f(x) \, dx\].
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Evaluated using \[F(b) - F(a)\].
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Gives a unique numerical value.
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Represents net area geometrically.
