Advertisements
Advertisements
Question
\[\int\frac{1}{a^2 x^2 + b^2} dx\]
Sum
Advertisements
Solution
\[\int\frac{dx}{a^2 x^2 + b^2}\]
\[ = \frac{1}{a^2}\int\frac{dx}{x^2 + \left( \frac{b}{a} \right)^2} \]
\[ = \frac{1}{a^2} \times \frac{a}{b} \tan^{- 1} \left( \frac{x}{\frac{b}{a}} \right) + C \left[ \therefore \int\frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{- 1} \left( \frac{x}{a} \right) + C \right]\]
\[ = \frac{1}{ab} \tan^{- 1} \left( \frac{ax}{b} \right) + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
` ∫ {sec x "cosec " x}/{log ( tan x) }` dx
\[\int\frac{- \sin x + 2 \cos x}{2 \sin x + \cos x} dx\]
\[\int\frac{1 + \cot x}{x + \log \sin x} dx\]
\[\int\frac{\sin \left( \tan^{- 1} x \right)}{1 + x^2} dx\]
\[\int {cosec}^4 \text{ 3x } \text{ dx } \]
\[\int \cot^n {cosec}^2 \text{ x dx } , n \neq - 1\]
\[\int \cos^7 x \text{ dx } \]
\[\int\frac{1}{4 x^2 + 12x + 5} dx\]
\[\int\frac{x}{x^4 - x^2 + 1} dx\]
\[\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)\]
\[\int\frac{1}{\sqrt{7 - 6x - x^2}} dx\]
\[\int\frac{1}{\sqrt{5 x^2 - 2x}} dx\]
\[\int\frac{x^2 + x + 1}{x^2 - x} dx\]
\[\int\frac{2}{2 + \sin 2x}\text{ dx }\]
\[\int\frac{1}{\cos 2x + 3 \sin^2 x} dx\]
\[\int\frac{1}{\sin x + \sqrt{3} \cos x} \text{ dx }\]
\[\int x \text{ sin 2x dx }\]
\[\int\left( x + 1 \right) \text{ e}^x \text{ log } \left( x e^x \right) dx\]
\[\int \cos^3 \sqrt{x}\ dx\]
\[\int e^x \cdot \frac{\sqrt{1 - x^2} \sin^{- 1} x + 1}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx\]
\[\int\frac{3 + 4x - x^2}{\left( x + 2 \right) \left( x - 1 \right)} dx\]
\[\int\frac{\sin 2x}{\left( 1 + \sin x \right) \left( 2 + \sin x \right)} dx\]
\[\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx\]
\[\int\frac{x^2}{\left( x^2 + 1 \right) \left( 3 x^2 + 4 \right)} dx\]
\[\int\frac{x^3 - 1}{x^3 + x} dx\]
Evaluate the following integral:
\[\int\frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)}dx\]
\[\int\frac{1}{\sin x \left( 3 + 2 \cos x \right)} dx\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} \text{ dx}\]
\[\int\frac{1}{\cos x + \sqrt{3} \sin x} \text{ dx } \] is equal to
\[\int\frac{1}{1 + \tan x} dx =\]
\[\int\frac{1}{1 - \cos x - \sin x} dx =\]
\[\int\frac{2}{\left( e^x + e^{- x} \right)^2} dx\]
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
\[\int\frac{1}{5 - 4 \sin x} \text{ dx }\]
\[\int x^3 \left( \log x \right)^2\text{ dx }\]
\[\int x\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx}\]
