Advertisements
Advertisements
Question
\[\int e^x \left( \cot x + \log \sin x \right) dx\]
Sum
Advertisements
Solution
\[\text{ Let I }= \int e^x \left( \cot x + \log \sin x \right)dx\]
\[\text{ Here,} f(x) = \log \sin x Put e^x f(x) = t\]
\[ \Rightarrow f'(x) = \cot x\]
\[\text{ let e}^x \log \sin x = t\]
\[\text{ Diff both sides w . r . t x}\]
\[ e^x \text{ log } \left( \sin x \right) + e^x \times \frac{1}{\sin x} \times \cos x = \frac{dt}{dx}\]
\[ \Rightarrow \left[ e^x \text{ log}\left( \sin x \right) + e^x \cot x \right]dx = dt\]
\[ \Rightarrow e^x \left( \cot x + \text{ log }\sin x \right)dx = dt\]
\[ \therefore \int e^x \left( \cot x + \log \sin x \right)dx = \int dt\]
\[ = t + C\]
\[ = e^x \log \sin x + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\left( 2 - 3x \right) \left( 3 + 2x \right) \left( 1 - 2x \right) dx\]
\[\int\left( x^e + e^x + e^e \right) dx\]
\[\int\frac{1 - \cos 2x}{1 + \cos 2x} dx\]
\[\int\frac{1}{\sqrt{2x + 3} + \sqrt{2x - 3}} dx\]
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
\[\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx\]
\[\int\frac{e^{3x}}{e^{3x} + 1} dx\]
` ∫ tan 2x tan 3x tan 5x dx `
\[\int\frac{\log\left( 1 + \frac{1}{x} \right)}{x \left( 1 + x \right)} dx\]
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
\[\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} dx\]
\[\int {cosec}^4 \text{ 3x } \text{ dx } \]
\[\int \sin^3 x \cos^6 x \text{ dx }\]
\[\int\frac{e^x}{\left( 1 + e^x \right)\left( 2 + e^x \right)} dx\]
\[\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
\[\int\frac{x}{\sqrt{8 + x - x^2}} dx\]
\[\int\frac{x + 2}{\sqrt{x^2 - 1}} \text{ dx }\]
\[\int\frac{x}{\sqrt{x^2 + x + 1}} \text{ dx }\]
\[\int\frac{1}{2 + \sin x + \cos x} \text{ dx }\]
\[\int x \sin x \cos x\ dx\]
\[\int x \sin^3 x\ dx\]
\[\int e^x \left( \log x + \frac{1}{x^2} \right) dx\]
\[\int\sqrt{3 - x^2} \text{ dx}\]
\[\int\left( 2x - 5 \right) \sqrt{2 + 3x - x^2} \text{ dx }\]
\[\int\frac{x^2 + 6x - 8}{x^3 - 4x} dx\]
\[\int\frac{5 x^2 + 20x + 6}{x^3 + 2 x^2 + x} dx\]
\[\int\frac{3x + 5}{x^3 - x^2 - x + 1} dx\]
\[\int\frac{1}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)} dx\]
\[\int e^x \left( 1 - \cot x + \cot^2 x \right) dx =\]
\[\int \cot^4 x\ dx\]
\[\int \cot^5 x\ dx\]
\[\int \sin^3 x \cos^4 x\ \text{ dx }\]
\[\int\frac{1}{\sqrt{x^2 - a^2}} \text{ dx }\]
\[\int\frac{1}{2 + \cos x} \text{ dx }\]
\[\int\frac{\sin^6 x}{\cos x} \text{ dx }\]
\[\int\sqrt{x^2 - a^2} \text{ dx}\]
\[\int\frac{\log \left( \log x \right)}{x} \text{ dx}\]
\[\int \sin^{- 1} \sqrt{x}\ dx\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
