Advertisements
Advertisements
Question
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
Sum
Advertisements
Solution
\[\int\left( \frac{1 + \cos 4x}{\cot x - \tan x} \right) dx\]
\[ = \int\frac{\left( 1 + \cos 4x \right)}{\left( \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \right)} dx\]
\[ = \int\frac{2 \cos^2 2x \times \sin x \cos x}{\left( \cos^2 x - \sin^2 x \right)}dx\]
\[ = \int\frac{\cos^2 2x \times 2 \sin x \cos x}{\cos 2x}dx\]
\[ = \int\cos 2x \sin 2xdx\]
\[ = \frac{1}{2}\int2 \sin 2x \cos 2xdx\]
\[ = \frac{1}{2}\int\sin 4xdx\]
\[ = \frac{1}{2}\left[ - \frac{\cos 4x}{4} \right] + C\]
\[ = - \frac{1}{8}\cos 4x + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
\[\int\frac{1 - \sin x}{x + \cos x} dx\]
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
\[\int\frac{\tan x}{\sqrt{\cos x}} dx\]
\[\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx\]
\[\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx\]
\[\int\frac{\cos^5 x}{\sin x} dx\]
` ∫ tan^5 x dx `
\[\int \sin^5 x \cos x \text{ dx }\]
\[\int\frac{1}{4 x^2 + 12x + 5} dx\]
\[\int\frac{\sec^2 x}{\sqrt{4 + \tan^2 x}} dx\]
\[\int\frac{x}{x^2 + 3x + 2} dx\]
\[\int\frac{2x}{2 + x - x^2} \text{ dx }\]
\[\int\frac{x^2 + x - 1}{x^2 + x - 6}\text{ dx }\]
\[\int\frac{\cos x}{\cos 3x} \text{ dx }\]
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int\frac{1}{p + q \tan x} \text{ dx }\]
\[\int\frac{5 \cos x + 6}{2 \cos x + \sin x + 3} \text{ dx }\]
\[\int x^2 \sin^2 x\ dx\]
\[\int \sec^{- 1} \sqrt{x}\ dx\]
\[\int\frac{x^2 \tan^{- 1} x}{1 + x^2} \text{ dx }\]
\[\int e^x \left[ \sec x + \log \left( \sec x + \tan x \right) \right] dx\]
\[\int e^x \left( \frac{\sin 4x - 4}{1 - \cos 4x} \right) dx\]
\[\int\frac{\sqrt{16 + \left( \log x \right)^2}}{x} \text{ dx}\]
\[\int\frac{x^2 + 1}{x\left( x^2 - 1 \right)} dx\]
\[\int\frac{x^2 + 6x - 8}{x^3 - 4x} dx\]
\[\int\frac{x^2 + 1}{\left( 2x + 1 \right) \left( x^2 - 1 \right)} dx\]
\[\int\frac{x}{4 + x^4} \text{ dx }\] is equal to
\[\int\sqrt{\frac{x}{1 - x}} dx\] is equal to
\[\int e^x \left\{ f\left( x \right) + f'\left( x \right) \right\} dx =\]
\[\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} \text{ dx }\]
\[\int\frac{1}{\text{ sin} \left( x - a \right) \text{ sin } \left( x - b \right)} \text{ dx }\]
\[\int \sin^3 x \cos^4 x\ \text{ dx }\]
\[\int \log_{10} x\ dx\]
\[\int \sin^{- 1} \sqrt{x}\ dx\]
\[\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int\frac{x^2}{\left( x - 1 \right)^3 \left( x + 1 \right)} \text{ dx}\]
\[\int \left( e^x + 1 \right)^2 e^x dx\]
