Advertisements
Advertisements
प्रश्न
\[\int\frac{\cos x}{\sin^2 x + 4 \sin x + 5} dx\]
बेरीज
Advertisements
उत्तर
` ∫ { cos x dx}/{sin^2 x + 4\sin x + 5}`
\[\text{ let }\sin x = t\]
\[ \Rightarrow \text{cos x dx }= dt\]
Now, ` ∫ { cos x dx}/{sin^2 x + 4\sin x + 5}`
\[ = \int\frac{dt}{t^2 + 4t + 5}\]
\[ = \int\frac{dt}{t^2 + 2 \times t \times 2 + 4 + 1}\]
\[ = \int\frac{dt}{\left( t + 2 \right)^2 + 1^2}\]
\[ = \frac{1}{1} \tan^{- 1} \left( \frac{t + 2}{1} \right) + C\]
\[ = \tan^{- 1} \left( \sin x + 2 \right) + C\]
shaalaa.com
या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{1}{1 - \cos x} dx\]
\[\int \left( a \tan x + b \cot x \right)^2 dx\]
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
\[\int\frac{1}{1 - \sin\frac{x}{2}} dx\]
\[\int \left( e^x + 1 \right)^2 e^x dx\]
\[\int \sin^2\text{ b x dx}\]
\[\int\frac{1}{ x \text{log x } \text{log }\left( \text{log x }\right)} dx\]
\[\int\frac{x + 1}{x \left( x + \log x \right)} dx\]
` ∫ tan 2x tan 3x tan 5x dx `
\[\int\frac{\sin 2x}{\sin \left( x - \frac{\pi}{6} \right) \sin \left( x + \frac{\pi}{6} \right)} dx\]
\[\int\frac{1}{1 + \sqrt{x}} dx\]
\[\int2x \sec^3 \left( x^2 + 3 \right) \tan \left( x^2 + 3 \right) dx\]
\[\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]
\[\int\frac{e^{m \tan^{- 1} x}}{1 + x^2} dx\]
\[\int\frac{x}{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}} dx\]
\[\int\frac{1}{x^2 \left( x^4 + 1 \right)^{3/4}} dx\]
\[\int\frac{\sin^5 x}{\cos^4 x} \text{ dx }\]
\[\int {cosec}^4 \text{ 3x } \text{ dx } \]
\[\int \sin^5 x \text{ dx }\]
\[\int\frac{1}{\sin^3 x \cos^5 x} dx\]
\[\int\frac{e^{3x}}{4 e^{6x} - 9} dx\]
\[\int\frac{e^x}{\left( 1 + e^x \right)\left( 2 + e^x \right)} dx\]
\[\int\frac{1}{\sqrt{5 - 4x - 2 x^2}} dx\]
\[\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)\]
` ∫ {x-3} /{ x^2 + 2x - 4 } dx `
\[\int\frac{1}{1 + 3 \sin^2 x} \text{ dx }\]
`int 1/(cos x - sin x)dx`
\[\int x e^x \text{ dx }\]
\[\int \sin^3 \sqrt{x}\ dx\]
\[\int e^x \cdot \frac{\sqrt{1 - x^2} \sin^{- 1} x + 1}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{1}{x \log x \left( 2 + \log x \right)} dx\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{2x + 3}} \text{ dx }\]
\[\int \sin^5 x\ dx\]
\[\int \cos^5 x\ dx\]
\[\int\sqrt{\frac{1 + x}{x}} \text{ dx }\]
\[\int\sqrt{1 + 2x - 3 x^2}\text{ dx } \]
\[\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }\]
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} \text{ dx}\]
Evaluate : \[\int\frac{\cos 2x + 2 \sin^2 x}{\cos^2 x}dx\] .
