Advertisements
Advertisements
Question
\[\int\frac{1}{\sqrt{1 - x^2}\left( 2 + 3 \sin^{- 1} x \right)} dx\]
Sum
Advertisements
Solution
\[\text{Let I} = \int\frac{1}{\sqrt{1 - x^2}\left( 2 + 3 \sin^{- 1} x \right)}dx\]
\[\text{Putting}\ \sin^{- 1} x = t\]
\[ \Rightarrow \frac{1}{\sqrt{1 - x^2}} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{1}{\sqrt{1 - x^2}}dx = dt\]
\[ \therefore I = \int\frac{1}{2 + 3t}dt\]
\[ = \frac{1}{3} \text{ln }\left| 2 + 3t \right| + C\]
\[ = \frac{1}{3} \text{ln }\left| 2 + 3 \sin^{- 1} x \right| + C \left[ \because t = \sin^{- 1} x \right]\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int \sin^{- 1} \left( \frac{2 \tan x}{1 + \tan^2 x} \right) dx\]
\[\int\frac{\cos x}{1 + \cos x} dx\]
\[\int\frac{x + 3}{\left( x + 1 \right)^4} dx\]
\[\int\left( 5x + 3 \right) \sqrt{2x - 1} dx\]
\[\int \sin^2\text{ b x dx}\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
` ∫ tan^5 x dx `
\[\int \cot^6 x \text{ dx }\]
\[\int \sin^4 x \cos^3 x \text{ dx }\]
\[\int\frac{1}{\sqrt{a^2 - b^2 x^2}} dx\]
\[\int\frac{e^x}{1 + e^{2x}} dx\]
` ∫ { x^2 dx}/{x^6 - a^6} dx `
\[\int\frac{\sin 2x}{\sqrt{\sin^4 x + 4 \sin^2 x - 2}} dx\]
\[\int\frac{\cos x - \sin x}{\sqrt{8 - \sin2x}}dx\]
\[\int\frac{x}{x^2 + 3x + 2} dx\]
\[\int\frac{x + 1}{\sqrt{x^2 + 1}} dx\]
\[\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{4 \sin x + 5 \cos x}{5 \sin x + 4 \cos x} \text{ dx }\]
\[\int\frac{x^2 \tan^{- 1} x}{1 + x^2} \text{ dx }\]
\[\int x \cos^3 x\ dx\]
\[\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx\]
\[\int\left( x - 2 \right) \sqrt{2 x^2 - 6x + 5} \text{ dx }\]
\[\int\left( 2x + 3 \right) \sqrt{x^2 + 4x + 3} \text{ dx }\]
\[\int\frac{x^2 + 1}{x^4 + x^2 + 1} \text{ dx }\]
\[\int\frac{\sin x}{1 + \sin x} \text{ dx }\]
\[\int\frac{1}{e^x + 1} \text{ dx }\]
\[\int\frac{5x + 7}{\sqrt{\left( x - 5 \right) \left( x - 4 \right)}} \text{ dx }\]
\[\int\frac{1}{\sin x \left( 2 + 3 \cos x \right)} \text{ dx }\]
\[\int\sqrt{\frac{a + x}{x}}dx\]
\[\int \tan^3 x\ \sec^4 x\ dx\]
\[\int\left( 2x + 3 \right) \sqrt{4 x^2 + 5x + 6} \text{ dx}\]
\[\int \left( x + 1 \right)^2 e^x \text{ dx }\]
\[\int\frac{\log \left( 1 - x \right)}{x^2} \text{ dx}\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }\]
\[\int\frac{1}{x\sqrt{1 + x^3}} \text{ dx}\]
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx}\]
\[\int\frac{\cot x + \cot^3 x}{1 + \cot^3 x} \text{ dx}\]
\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]
