Advertisements
Advertisements
Question
\[\int \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx\]
Sum
Advertisements
Solution
\[\text{ Let I } = \int \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) \text{ dx }\]
\[ = \int3 \tan^{- 1} \left( x \right) \text{ dx }\]
\[ = 3\int\left[ \tan^{- 1} \left( x \right) \times 1 \right] \text{ dx }\]
\[ = 3 \left[ \tan^{- 1} x \times x - \int\frac{1}{1 + x^2} \times\text{ x dx } \right]\]
\[ = 3x \tan^{- 1} x - 3\int\frac{x}{1 + x^2} dx\]
\[\text{ let 1 }+ x^2 = t\]
\[ \Rightarrow \text{ 2x dx }= dt\]
\[\text{ Then,} \]
\[I = 3x \tan^{- 1} x - \frac{3}{2}\int\frac{dt}{t}\]
\[ = 3x \tan^{- 1} x - \frac{3}{2} \text{ log } \left| t \right| + C\]
\[ = 3x \tan^{- 1} x - \frac{3}{2} \text{ log} \left| 1 + x^2 \right| + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx\]
\[\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx\]
\[\int\frac{1}{\text{cos}^2\text{ x }\left( 1 - \text{tan x} \right)^2} dx\]
\[\int\frac{3x + 5}{\sqrt{7x + 9}} dx\]
\[\int\frac{\text{sin} \left( x - a \right)}{\text{sin}\left( x - b \right)} dx\]
\[\int\frac{\sin 2x}{\sin \left( x - \frac{\pi}{6} \right) \sin \left( x + \frac{\pi}{6} \right)} dx\]
\[\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx\]
\[\int\frac{1}{\sqrt{x} + x} \text{ dx }\]
\[\int\frac{1}{x^2 \left( x^4 + 1 \right)^{3/4}} dx\]
\[\int\frac{x + 1}{x^2 + x + 3} dx\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
\[\int\frac{x^2 \left( x^4 + 4 \right)}{x^2 + 4} \text{ dx }\]
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
\[\int\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int\frac{8 \cot x + 1}{3 \cot x + 2} \text{ dx }\]
`int"x"^"n"."log" "x" "dx"`
\[\int e^\sqrt{x} \text{ dx }\]
\[\int e^x \left( \tan x - \log \cos x \right) dx\]
\[\int e^x \left( \cot x + \log \sin x \right) dx\]
\[\int e^x \frac{\left( 1 - x \right)^2}{\left( 1 + x^2 \right)^2} \text{ dx }\]
∴\[\int e^{2x} \left( - \sin x + 2 \cos x \right) dx\]
\[\int e^x \left( \frac{\sin x \cos x - 1}{\sin^2 x} \right) dx\]
\[\int\sqrt{2ax - x^2} \text{ dx}\]
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
\[\int\frac{1}{1 + x + x^2 + x^3} dx\]
\[\int\frac{x^4}{\left( x - 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }\]
\[\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 1}} \text{ dx }\]
Write the anti-derivative of \[\left( 3\sqrt{x} + \frac{1}{\sqrt{x}} \right) .\]
\[\int e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx\]
\[\int\frac{\sin^2 x}{\cos^4 x} dx =\]
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int \cot^4 x\ dx\]
\[\int \cos^5 x\ dx\]
\[\int\frac{1}{4 x^2 + 4x + 5} dx\]
\[\int\frac{1}{\sin x \left( 2 + 3 \cos x \right)} \text{ dx }\]
\[\int\frac{6x + 5}{\sqrt{6 + x - 2 x^2}} \text{ dx}\]
\[\int x^3 \left( \log x \right)^2\text{ dx }\]
\[\int \sin^{- 1} \sqrt{x}\ dx\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
