# 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**

- \(\large \int \; dx = x + C\)
- \(\large \int a \; dx = ax + C\)
- \(\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1\)
- \(\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1\)

**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

∫ sec^{2}x dx = tan x + c

∫ cosec^{2}x 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^{-1}x dx = x sin^{-1}x + \(\sqrt{1-{x}^2}\) + c

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

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

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

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

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

**Integral formulas for exponential trigonometric functions:**

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

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

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

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

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

∫ sin^{2}x dx = \(\frac{1}{2}\)x – \(\frac{1}{4}\)sin 2x + c

∫ cos^{2}x dx = \(\frac{1}{2}\)x + \(\frac{1}{4}\)sin 2x + c

∫ tan^{2}x dx = tan x – x + c

∫ cot^{2}x dx = -cot x – x + c

**Integral formulas for logarithmic functions:**

∫ a^{x} 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 x^{2} +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 2 ^{nd} function i.e. u and v:**

**I – Inverse Function**

**L – Logarithmic Function**

**A – Algebraic Function**

**T- Trigonometric Function**

**E- Exponential Function**

∫ e^{x} [f(x) + f’(x)] dx = ∫e^{x} 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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