Advertisements
Advertisements
Question
\[\int {cosec}^4 \text{ 3x } \text{ dx } \]
Sum
Advertisements
Solution
∫ cosec4 3x dx
= ∫ cosec2 3x . cosec2 3x dx
= ∫ (1 + cot2 3x) cosec2 3x dx
Let cot (3x) = t
⇒ –cosec2 (3x) × 3 dx = dt
\[\Rightarrow {cosec}^2 \left( 3x \right)dx = - \frac{dt}{3}\]
\[Now, \int\left( 1 + \cot^2 3x \right)\]
\[ = \frac{- 1}{3}\int\left( 1 + t^2 \right) dt\]
\[ = \frac{- 1}{3} \left[ t + \frac{t^3}{3} \right] + C\]
\[ = - \frac{t}{3} - \frac{t^3}{9} + C\]
\[ = \frac{- \text{cot} \left( 3x \right)}{3} - \frac{\text{ cot }^3 3x}{9} + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\sqrt{x}\left( 3 - 5x \right) dx\]
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
\[\int\frac{1}{2 - 3x} + \frac{1}{\sqrt{3x - 2}} dx\]
\[\int\frac{x + 3}{\left( x + 1 \right)^4} dx\]
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
\[\int \tan^2 \left( 2x - 3 \right) dx\]
\[\int\frac{\cos x}{\cos \left( x - a \right)} dx\]
\[\int\frac{\tan x}{\sqrt{\cos x}} dx\]
\[\int2x \sec^3 \left( x^2 + 3 \right) \tan \left( x^2 + 3 \right) dx\]
\[\int\frac{\cos^5 x}{\sin x} dx\]
\[\int\frac{\cos\sqrt{x}}{\sqrt{x}} dx\]
\[\int\frac{1}{\left( x + 1 \right)\left( x^2 + 2x + 2 \right)} dx\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
\[\int \cos^5 x \text{ dx }\]
\[\int\frac{\sec^2 x}{1 - \tan^2 x} dx\]
\[\int\frac{1}{\sqrt{2x - x^2}} dx\]
\[\int\frac{\sin 2x}{\sqrt{\cos^4 x - \sin^2 x + 2}} dx\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int x \text{ sin 2x dx }\]
\[\int \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ dx }\]
\[\int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx\]
\[\int\frac{2x - 3}{\left( x^2 - 1 \right) \left( 2x + 3 \right)} dx\]
\[\int\frac{1}{x \log x \left( 2 + \log x \right)} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x + 1 \right)^2} dx\]
\[\int\frac{\cos x}{\left( 1 - \sin x \right)^3 \left( 2 + \sin x \right)} dx\]
\[\int\frac{x^2 + 1}{x^4 + 7 x^2 + 1} 2 \text{ dx }\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{x^2 + 1}} \text{ dx }\]
\[\int \sec^2 x \cos^2 2x \text{ dx }\]
\[\int\frac{1}{e^x + 1} \text{ dx }\]
\[\int\frac{1}{e^x + e^{- x}} dx\]
\[\int\frac{x^2}{\left( x - 1 \right)^3} dx\]
\[\int x \sin^5 x^2 \cos x^2 dx\]
\[\int\frac{1}{3 x^2 + 13x - 10} \text{ dx }\]
\[\int\frac{1}{a + b \tan x} \text{ dx }\]
\[\int\frac{1}{1 - 2 \sin x} \text{ dx }\]
\[\int \tan^3 x\ \sec^4 x\ dx\]
\[\int\sqrt{a^2 - x^2}\text{ dx }\]
\[\int e^{2x} \left( \frac{1 + \sin 2x}{1 + \cos 2x} \right) dx\]
