Advertisements
Advertisements
प्रश्न
\[\int\text{sin mx }\text{cos nx dx m }\neq n\]
बेरीज
Advertisements
उत्तर
\[\int\text{sin }\left( mx \right) \cdot \text{cos} \left( nx \right) dx\]
\[ = \frac{1}{2}\int2 \text{sin} \left( mx \right) \cdot \text{cos} \left( nx \right)dx\]
\[ = \frac{1}{2}\int\left[ \text{sin} \left( mx + nx \right) + \text{sin} \left( mx - nx \right) \right]dx \left[ \therefore \text{2 sin A }\cdot \text{cos B} = \text{sin} \left( A + B \right) + \text{sin} \left( A - B \right) \right]\]
\[ = \frac{1}{2}\left[ - \frac{\text{cos} \left( m + n \right)x}{m + n} - \frac{\text{cos} \left( m - n \right)x}{m - n} \right] + C\]
shaalaa.com
या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{1 - \cos 2x}{1 + \cos 2x} dx\]
If f' (x) = a sin x + b cos x and f' (0) = 4, f(0) = 3, f
\[\left( \frac{\pi}{2} \right)\] = 5, find f(x)
\[\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx\]
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
Integrate the following integrals:
\[\int\text { sin x cos 2x sin 3x dx}\]
\[\int\sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}} dx\]
\[\int\sqrt{\frac{1 - \sin 2x}{1 + \sin 2x}} dx\]
\[\int\frac{2 \cos 2x + \sec^2 x}{\sin 2x + \tan x - 5} dx\]
` ∫ tan 2x tan 3x tan 5x dx `
\[\int\frac{e^{m \tan^{- 1} x}}{1 + x^2} dx\]
\[\int \sin^5 x \text{ dx }\]
\[\int x \cos^3 x^2 \sin x^2 \text{ dx }\]
\[\int\frac{1}{4 x^2 + 12x + 5} dx\]
\[\int\frac{x}{x^4 + 2 x^2 + 3} dx\]
` ∫ { x^2 dx}/{x^6 - a^6} dx `
\[\int\frac{2x + 5}{x^2 - x - 2} \text{ dx }\]
\[\int\frac{x}{\sqrt{x^2 + 6x + 10}} \text{ dx }\]
\[\int\frac{5 \cos x + 6}{2 \cos x + \sin x + 3} \text{ dx }\]
\[\int x^2 \text{ cos x dx }\]
\[\int \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx\]
\[\int x^2 \tan^{- 1} x\text{ dx }\]
\[\int x \sin x \cos 2x\ dx\]
\[\int e^x \left( \cos x - \sin x \right) dx\]
\[\int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx\]
\[\int\frac{\sqrt{16 + \left( \log x \right)^2}}{x} \text{ dx}\]
\[\int\frac{5x}{\left( x + 1 \right) \left( x^2 - 4 \right)} dx\]
\[\int\frac{x^2 + 6x - 8}{x^3 - 4x} dx\]
\[\int\frac{18}{\left( x + 2 \right) \left( x^2 + 4 \right)} dx\]
\[\int\frac{x^3 - 1}{x^3 + x} dx\]
\[\int\frac{1}{x \left( x^4 - 1 \right)} dx\]
\[\int\frac{1}{x^4 - 1} dx\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{2x + 3}} \text{ dx }\]
\[\int\frac{x^2}{\left( x - 1 \right) \sqrt{x + 2}}\text{ dx}\]
\[\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 9}} \text{ dx}\]
\[\int\frac{2}{\left( e^x + e^{- x} \right)^2} dx\]
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
\[\int \left( \sin^{- 1} x \right)^3 dx\]
\[\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }\]
\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]
\[\int\frac{x^2}{x^2 + 7x + 10}\text{ dx }\]
