Integrals

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
    \int \frac{quadratic}{Bi quadratic} dx (or) \int \frac{1}{Bi quadratic} dx
  • 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

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  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