# Introduction to Integral formulas:

The list of integral calculus formula is here with all the rules which are needed to solve integration. The formula sheet of integration include basic integral formulas, integration by parts and partial fraction, area as a sum and properties of definite integral. At first take a look at indefinite integration.

## Indefinite integral formulas:

• Integration is the inverses of differentiation.
• Two indefinite integrals having same derivative lead to the same family of curves, this makes them equivalent.
• Addition rule of integration: ∫ [ f(x) + g(x) ]dx = ∫f(x) dx + ∫g(x) dx
• The multiplication rule for any real number k, ∫k f(x) dx = k ∫f(x) dx
1. $$\large \int \; dx = x + C$$
2. $$\large \int a \; dx = ax + C$$
3. $$\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1$$
4. $$\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1$$

## Integral formulas of trigonometric functions:

∫ sin x dx = -cos x + c

∫ cos x dx = sin x + c

∫ tan x dx = log|sec x| + c

∫ cot x dx = log|sin x| + c

∫ sec x dx = log|sec x + tan x| + c

∫ cosec x dx = log|cosec x – cot x| + c

∫ sec2x dx = tan x + c

∫ cosec2x dx = -cot x + c

∫ sec x tan x dx = sec x + c

∫ cosec x cot x dx = – cosec x + c

## Integral formulas for inverse trigonometric functions:

$$\large \int \frac{1}{\sqrt{1-x^{2}}} \; dx = \sin^{-1} x + C$$

$$\large \int \frac{1}{\sqrt{1+x^{2}}} \; dx = \tan^{-1} x + C$$

$$\large \int \frac{1}{|x|\sqrt{x^{2}-1}} \; dx = \sec^{-1} x + C$$

$$\large \int \frac{dx}{\sqrt{x^{2}+ {a}^2}} \; dx = \frac{1}{a}\;{tan}^{-1} \frac{x}{a} + C$$

$$\large \int \frac{dx}{\sqrt{a^{2}- {x}^2}} \; dx = {sin}^{-1} \frac{x}{a} + C$$

∫ sin-1x dx = x sin-1x + $$\sqrt{1-{x}^2}$$ + c

∫ cos-1x dx = x cos-1x – $$\sqrt{1-{x}^2}$$ + c

∫ tan-1x dx = x tan-1x – ln$$\sqrt{1 + {x}^2}$$ + c

∫ cot-1x dx = x cot-1x +ln$$\sqrt{1 + {x}^2}$$ + c

∫ sec-1x dx = x sec-1x – ln|x + $$\sqrt{|{x}^2-1|}$$| + c

∫ cosec-1x dx = x cosec-1x + ln|x + $$\sqrt{|{x}^2-1|}$$| + c

## Integral formulas for exponential trigonometric functions:

$$\large \int \sin^{n} (x) dx = \frac{-1}{n} \sin^{n-1} (x) \cos (x) +\frac{n-1}{n} \int \sin^{n-2} (x) dx$$

$$\large \int \cos^{n} (x) dx = \frac{1}{n} \cos^{n-1} (x) \sin (x) +\frac{n-1}{n} \int \cos^{n-2} (x) dx$$

$$\large \int \tan^{n} (x) dx = \frac{1}{n-1} \tan^{n-1} (x) +\int \tan^{n-2} (x) dx$$

$$\large \int \sec^{n} (x) dx = \frac{1}{n-1} \sec^{n-2} (x) \tan (x) + \frac{n-2}{n-1} \int \sec^{n-2} (x) dx$$

$$\large \int \csc^{n} (x) dx = \frac{-1}{n-1} \csc^{n-2} (x) \cot (x) + \frac{n-2}{n-1} \int \csc^{n-2} (x) dx$$

∫ sin2x dx = $$\frac{1}{2}$$x – $$\frac{1}{4}$$sin 2x + c

∫ cos2x dx = $$\frac{1}{2}$$x + $$\frac{1}{4}$$sin 2x + c

∫ tan2x dx = tan x – x + c

∫ cot2x dx = -cot x – x + c

## Integral formulas for logarithmic functions:

∫ ax dx = $$\frac{{a}^x}{log\:a}$$ + c

∫ $$\frac{1}{x}$$ dx = log|x| + c

∫ $$\frac{dx}{{x}^2 –{a}^2}$$ =$$\frac{1}{2a}$$ log|$$\frac{x\:-\:a}{x\:+\:a}$$| + c

∫$$\frac{dx}{{a}^2 –{x}^2}$$ = $$\frac{1}{2a}$$ log|$$\frac{a\:+\:x}{a\:-\:x}$$| + c

∫$$\frac{dx}{\sqrt{{x}^2 –{a}^2}}$$ = log |x + $$\sqrt{{x}^2 – {a}^2}$$| + c

∫$$\frac{dx}{\sqrt{{x}^2 +{a}^2}}$$ = log |x + $$\sqrt{{x}^2 + {a}^2}$$| + c

## Integral formulas for partial fractions:

$$\frac{px\:+\:q}{(x-a)(x-b)}$$ = $$\frac{A}{x\:-\:a} + \frac{B}{x\:-\:b}$$ , a ≠ b

$$\frac{px\:+\:q}{(x-a)^{2}}$$ = $$\frac{A}{x\:-\:a} + \frac{B}{(x-a)^{2}}$$

$$\frac{p{x}^2+qx+r}{(x-a)(x-b)(x-c)}$$ = $$\frac{A}{x\:-\:a} + \frac{B}{x\:-\:b} + \frac{C}{x\:-\:c}$$

$$\frac{p{x}^2+qx+r}{(x-a)^{2}(x-b)}$$ = $$\frac{A}{x\:-\:a} + \frac{B}{(x-a)^{2}} + \frac{C}{x-b}$$

$$\frac{p{x}^2+qx+r}{(x-a) ({x}^2 + bx +c)}$$ = $$\frac{A}{x-a} + {Bx+C}{{x}^2 + bx +c}$$

Where x2 +bx +c can’t be factorized further.

## Integral Formulas by parts:

$$\large \int u\;v\;dx=u\int v\;dx-\int\left(\frac{du}{dx}\int v\;dx\right)dx$$

In this formula there is simple rule to know first and 2nd function i.e. u and v:

I – Inverse Function

L – Logarithmic Function

A – Algebraic Function

T- Trigonometric Function

E- Exponential Function

∫ ex [f(x) + f’(x)] dx = ∫ex f(x) dx + c

## Some Special integral formulas:

∫ $$\sqrt{{x}^2 – {a}^2}$$ dx = $$\frac{x}{2}\;\sqrt{{x}^2 – {a}^2}$$ – $$\frac{{a}^2}{2}$$log|x + $$\sqrt{{x}^2 – {a}^2}$$| + c

∫$$\sqrt{{x}^2 + {a}^2}$$ dx = $$\frac{x}{2}\;\sqrt{{x}^2 + {a}^2}$$ + $$\frac{{a}^2}{2}$$log|x + $$\sqrt{{x}^2 + {a}^2}$$| + c

∫$$\sqrt{{a}^2 – {x}^2}$$ dx = $$\frac{x}{2}\;\sqrt{{x}^2 – {a}^2}$$ + $$\frac{{a}^2}{2}$$ sin-1$$\frac{x}{a}$$ + c

## Definite integral formulas:

### Actual Rule:

$$\large \int_{a}^{\infty}f(x)dx=\lim_{b\rightarrow \infty}\left [ \int_{a}^{b}f(x)dx\right ]$$

Or we can write:

$$\large \int_{a}^{b}f(x)dx=F(b)-F(a)$$

Here a is lower limit and b, ∞ are upper limits.

## Some rational formula for definite integration:

$$\large \int_{a}^{\infty }\frac{dx}{x^{2}+a^{2}}=\frac{\pi }{2a}$$

$$\large \int_{a}^{\infty }\frac{x^{m}dx}{x^{n}+a^{n}}=\frac{\pi a^{m-n+1}}{n\sin \left ( \frac{(m+1)\pi }{n} \right )},0< m+1< n$$\)

$$\large \int_{a}^{\infty }\frac{x^{p-1}dx}{1+x}=\frac{\pi }{\sin (p\pi )},0< p< 1$$

$$\large \int_{a}^{\infty }\frac{x^{m}dx}{1+2x\cos \beta +x^{2}}=\frac{\pi \sin (m\beta )}{\sin (m\pi )\sin \beta }$$

$$\large \int_{a}^{\infty }\frac{dx}{\sqrt{a^{2}-x^{2}}}=\frac{\pi }{2}$$

$$\large \int_{a}^{\infty }\sqrt{a^{2}-x^{2}}dx=\frac{\pi a^{2}}{4}$$

## Definite integration formula for trigonometric functions:

$$\large \int_{0}^{\pi }\sin(mx)\sin (nx)dx=\left\{\begin{matrix} 0 & if\;m\neq n\\ \frac{\pi }{2} & if\;m=n \end{matrix}\right.\;m,n\;positive\;integers$$

$$\large \int_{0}^{\pi }\cos (mx)\cos (nx)dx=\left\{\begin{matrix} 0 & if\;m\neq n\\ \frac{\pi }{2} & if\;m=n \end{matrix}\right.\;m,n\;positive\;integers$$

$$\large \int_{0}^{\pi }\sin (mx)\cos (nx)dx=\left\{\begin{matrix} 0 & if\;m+n\;even\\ \frac{2m}{m^{2}-n^{2}} & if\;m+n\;odd \end{matrix}\right.\;m,n\;integers$$

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