Integrals of some Particular Functions

TIPS FOR SOLVING INTEGRATING FUNCTIONS

• Integrals are in the form
\int \frac{1}{ax^{2} + bx + c} dx (or) \int \frac{1}{\sqrt{ax^{2} + bx + c}} dx (or) \int \sqrt{ax^{2} + bx + c}  dx
• Process : Converting ax2 + bx + c into perfect square which is in the form a2 + x2, a2 − x2 (or) x2 − a2.
• Integrals are in the form
\int \frac{px + q}{ax^{2} + bx + c} dx (or) \int \frac{px + q}{\sqrt{ax^{2} + bx + c}} dx (or) \int (px + q) \sqrt{ax^{2} + bx + c}  dx
• Process :
Put px + q =A \frac{d}{dx} (ax^{2} + bx + c) + B
Find the values of A and B by comparing coefficients of u and constant terms.
• Integrals which are in the form
\int \frac{f(x)}{g(x)} dx
(Where degree of f(x) ≥ degree of g(x))
• Process :
Divide the numerator with the denominator upto the level of degree of remainder must be less than the degree of divisor then write the given integral as I = \int \left(\phi + \frac{R}{D}\right) dx
Where = Quotient
= Remainder
D = Divisor
• Integrals which are in the form
• Process :
Divide the Numerator and denominator with x2 and then use substitution method

IMPORTANT

• \int \frac{1}{x^{2} - a^{2}} dx = \frac{1}{2a} \log \begin{vmatrix}\frac{x - a}{x+a} \end{vmatrix} + c
\int \frac{1}{a^{2} - x^{2}} dx = \frac{1}{2a} \log \begin{vmatrix}\frac{a + x}{a-x} \end{vmatrix} + c
• \int \frac{1}{a^{2}+x^{2}} dx = \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + c
• \int \frac{1}{\sqrt{a^{2}-x^{2}}} dx = \sin^{-1} \left(\frac{x}{a}\right) + c
• \int \frac{1}{\sqrt{x^{2} - a^{2}}} dx = \log \begin{vmatrix} x + \sqrt{x^{2} - a^{2}}\end{vmatrix} + c
\int \frac{1}{\sqrt{x^{2} + a^{2}}} dx = \log \begin{vmatrix} x + \sqrt{x^{2} + a^{2}}\end{vmatrix} + c

View the Topic in this video From 01:45 To 47:09

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

1. \int \frac{1}{x^{2}+a^{2}}dx= \frac{1}{a}\tan^{-1} \frac{x}{a} + C
2. \int \frac{1}{a^{2}-x^{2}}dx= \frac{1}{2a}\log\begin{vmatrix}\frac{a+x}{a-x}\end{vmatrix}+ C=\frac{1}{a} \tan h^{-1}\left(\frac{x}{a}\right)+C
3. \int \frac{dx}{x^{2}-a^{2}}dx= \frac{1}{2a}\log\begin{vmatrix}\frac{x-a}{x+a}\end{vmatrix}+ C=-\frac{1}{a} \cot h^{-1}\left(\frac{x}{a}\right)+C
4. \int \frac{dx}{\sqrt{a^{2}-x^{2}}}= \sin^{-1}\frac{x}{a}+ C
5. \int \frac{dx}{\sqrt{x^{2}-a^{2}}}= \log\begin{vmatrix}x+\sqrt{x^{2}-a^{2}}\end{vmatrix}+ C=\cos h^{-1}\left(\frac{x}{a}\right)+C
6. \int \frac{dx}{\sqrt{x^{2}+a^{2}}}= \log\begin{vmatrix}x+\sqrt{x^{2}+a^{2}}\end{vmatrix}+ C=\sin h^{-1}\left(\frac{x}{a}\right)+C
7. \int \sqrt{x^{2}+a^{2}} \ dx= \frac{1}{2}\left[x\sqrt{x^{2}+a^{2}}+a^{2} \log\begin{vmatrix}x+\sqrt{x^{2}+a^{2}}\end{vmatrix}\right]+ C
8. \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
9. \int \sqrt{x^{2}-a^{2}} \ dx= \frac{1}{2}\left[x\sqrt{x^{2}-a^{2}}-a^{2} \log\begin{vmatrix}x+\sqrt{x^{2}-a^{2}}\end{vmatrix}\right]+ C

TRICKS ON INTEGRALS

• \int \frac{f'{x}}{\sqrt{a^{2} - \left(f(x)\right)^{2}}} dx = \sin^{-1} \left(\frac{f(x)}{a}\right) + c
• \int \frac{f'{x}}{\sqrt{\left(f(x)\right)^{2} - a^{2}}} dx = \log \begin{vmatrix} f(x) + \sqrt{\left(f(x)\right)^{2} - a^{2}} \end{vmatrix} + c
• \int \frac{f'{x}}{\sqrt{\left(f(x)\right)^{2} + a^{2}}} dx = \log \begin{vmatrix} f(x) + \sqrt{\left(f(x)\right)^{2} + a^{2}} \end{vmatrix} + c
• \int \frac{1}{(x+a)(x+b)} dx = \frac{1}{b-a} \log \begin{vmatrix}\frac{x+a}{x+b} \end{vmatrix}+ c
• \int \frac{1}{(ax+b)(cx+d)} dx = \frac{1}{ad-bc} \log \begin{vmatrix}\frac{ax+b}{cx+d} \end{vmatrix}+ c
• \int \frac{1}{\left(x^{2} + a^{2}\right) \left(x^{2} + b^{2}\right)} dx = \frac{1}{b^{2} - a^{2}} \left[\frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) - \frac{1}{b} \tan^{-1} \left(\frac{x}{b}\right)\right] + c
• \int \frac{x}{\left(x^{2} + a^{2}\right) \left(x^{2} + b^{2}\right)} dx = \frac{1}{2 \left(b^{2} - a^{2}\right)} \log \begin{vmatrix} \frac{x^{2} + b^{2}}{x^{2} + a^{2}} \end{vmatrix} + c
• \int \frac{1}{x(x^{n} + 1)} dx = \frac{1}{n} \log \begin{vmatrix} \frac{x^{n}}{1 + x^{n}} \end{vmatrix} + c
• \int \frac{1}{x(x^{n} - 1)} dx = \frac{1}{n} \log \begin{vmatrix} \frac{x^{n}-1}{x^{n}} \end{vmatrix} + c
• \int \frac{1}{x (1-x^{2})} dx = \frac{1}{n} \log \begin{vmatrix} \frac{x^{n}}{1-x^{n}} \end{vmatrix} + c