Advertisements
Advertisements
Question
\[\int\text{sin mx }\text{cos nx dx m }\neq n\]
Sum
Solution
\[\int\text{sin }\left( mx \right) \cdot \text{cos} \left( nx \right) dx\]
\[ = \frac{1}{2}\int2 \text{sin} \left( mx \right) \cdot \text{cos} \left( nx \right)dx\]
\[ = \frac{1}{2}\int\left[ \text{sin} \left( mx + nx \right) + \text{sin} \left( mx - nx \right) \right]dx \left[ \therefore \text{2 sin A }\cdot \text{cos B} = \text{sin} \left( A + B \right) + \text{sin} \left( A - B \right) \right]\]
\[ = \frac{1}{2}\left[ - \frac{\text{cos} \left( m + n \right)x}{m + n} - \frac{\text{cos} \left( m - n \right)x}{m - n} \right] + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{5 \cos^3 x + 6 \sin^3 x}{2 \sin^2 x \cos^2 x} dx\]
\[\int\frac{\cos x}{1 - \cos x} \text{dx }or \int\frac{\cot x}{\text{cosec } {x }- \cot x} dx\]
\[\int\frac{\cos^2 x - \sin^2 x}{\sqrt{1} + \cos 4x} dx\]
\[\int \left( a \tan x + b \cot x \right)^2 dx\]
\[\int\frac{1 + \cos x}{1 - \cos x} dx\]
\[\int\frac{x^3}{x - 2} dx\]
` ∫ {sin 2x} /{a cos^2 x + b sin^2 x } ` dx
\[\int\frac{\cos 4x - \cos 2x}{\sin 4x - \sin 2x} dx\]
\[\int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x e^x \right)} dx\]
\[\int\frac{x^2}{\sqrt{x - 1}} dx\]
` ∫ tan^3 x sec^2 x dx `
` ∫ tan^5 x dx `
` ∫ \sqrt{tan x} sec^4 x dx `
\[\int \sin^5 x \text{ dx }\]
\[\int \sin^7 x \text{ dx }\]
\[\int\frac{1}{\sqrt{1 + 4 x^2}} dx\]
\[\int\frac{1}{x^2 + 6x + 13} dx\]
\[\int\frac{dx}{e^x + e^{- x}}\]
\[\int\frac{3 x^5}{1 + x^{12}} dx\]
\[\int\frac{1}{\sqrt{3 x^2 + 5x + 7}} dx\]
\[\int\frac{x - 1}{3 x^2 - 4x + 3} dx\]
\[\int\frac{x^2 + x - 1}{x^2 + x - 6}\text{ dx }\]
\[\int\frac{x}{\sqrt{x^2 + 6x + 10}} \text{ dx }\]
\[\int\frac{x + 2}{\sqrt{x^2 - 1}} \text{ dx }\]
\[\int\frac{1}{\left( \sin x - 2 \cos x \right)\left( 2 \sin x + \cos x \right)} \text{ dx }\]
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int\frac{1}{p + q \tan x} \text{ dx }\]
\[\int x^3 \cos x^2 dx\]
\[\int\frac{x^2 \tan^{- 1} x}{1 + x^2} \text{ dx }\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{\left( x^2 + 1 \right) \left( x^2 + 2 \right)}{\left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx\]
\[\int\frac{1}{\left( x + 1 \right) \sqrt{x^2 + x + 1}} \text{ dx }\]
\[\int\frac{e^x - 1}{e^x + 1} \text{ dx}\]
\[\int \cos^5 x\ dx\]
\[\int\frac{1}{2 - 3 \cos 2x} \text{ dx }\]
\[\int\frac{\cos x}{\frac{1}{4} - \cos^2 x} \text{ dx }\]
\[\int \log_{10} x\ dx\]
\[\int x^3 \left( \log x \right)^2\text{ dx }\]
\[\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx}\]
\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]
