Advertisements
Advertisements
प्रश्न
\[\int\frac{- \sin x + 2 \cos x}{2 \sin x + \cos x} dx\]
योग
Advertisements
उत्तर
\[\text{Let I} = \int\frac{- \sin x + 2\cos x}{2\sin x + \cos x}dx\]
\[\text{Putting}\ 2\sin x + \cos x = t\]
\[ \Rightarrow 2\cos x - \sin x = \frac{dt}{dx}\]
\[ \Rightarrow \left( - \sin x + 2\cos x \right)dx = dt\]
\[ \therefore I = \int\frac{1}{t}dt\]
\[ = \text{ln}\left| t \right| + C\]
\[ = \text{ln }\left| 2\sin x + \cos x \right| + C \left[ \because t = 2\sin x + \cos x \right]\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
` ∫ {cosec x} / {"cosec x "- cot x} ` dx
\[\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx\]
\[\int\frac{1}{\left( 7x - 5 \right)^3} + \frac{1}{\sqrt{5x - 4}} dx\]
\[\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx\]
\[\int\frac{x^2 + 5x + 2}{x + 2} dx\]
` ∫ cos mx cos nx dx `
Integrate the following integrals:
\[\int\text{sin 2x sin 4x sin 6x dx} \]
\[\int\frac{1 - \sin x}{x + \cos x} dx\]
\[\int\frac{1 - \sin 2x}{x + \cos^2 x} dx\]
\[\int\frac{1}{1 + \sqrt{x}} dx\]
\[\int\frac{\cos^5 x}{\sin x} dx\]
\[\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]
\[\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }\]
\[\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx\]
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 + 1}} dx\]
\[\int\frac{1}{\sqrt{7 - 6x - x^2}} dx\]
\[\int\frac{\sin 2x}{\sqrt{\cos^4 x - \sin^2 x + 2}} dx\]
` ∫ {x-3} /{ x^2 + 2x - 4 } dx `
\[\int\frac{x + 1}{\sqrt{4 + 5x - x^2}} \text{ dx }\]
\[\int\frac{1}{4 \sin^2 x + 5 \cos^2 x} \text{ dx }\]
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
\[\int\frac{1}{p + q \tan x} \text{ dx }\]
\[\int e^x \frac{x - 1}{\left( x + 1 \right)^3} \text{ dx }\]
\[\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }\]
\[\int\left( x - 2 \right) \sqrt{2 x^2 - 6x + 5} \text{ dx }\]
\[\int\frac{5x}{\left( x + 1 \right) \left( x^2 - 4 \right)} dx\]
\[\int\frac{1}{x\left( x^n + 1 \right)} dx\]
\[\int\frac{1}{x \left( x^4 + 1 \right)} dx\]
\[\int\frac{1}{x \left( x^4 - 1 \right)} dx\]
\[\int\frac{1}{\sin x + \sin 2x} dx\]
\[\int\frac{x^4}{\left( x - 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \sqrt{x + 2}}\text{ dx}\]
\[\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 1}} \text{ dx }\]
\[\int\frac{\sin^6 x}{\cos^8 x} dx =\]
\[\int\frac{1}{3 x^2 + 13x - 10} \text{ dx }\]
\[\int x\sqrt{1 + x - x^2}\text{ dx }\]
\[\int\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} \text{ dx}\]
\[\int\frac{\cos^7 x}{\sin x} dx\]
