Advertisements
Advertisements
प्रश्न
\[\int e^{2x} \left( \frac{1 + \sin 2x}{1 + \cos 2x} \right) dx\]
योग
Advertisements
उत्तर
\[\text{We have}, \]
\[I = \int e^{2x} \left( \frac{1 + \sin 2x}{1 + \cos 2x} \right)dx\]
\[ = \int e^{2x} \left( \frac{1}{1 + \cos 2x} + \frac{\sin 2x}{1 + \cos 2x} \right)dx\]
\[ = \int e^{2x} \left( \frac{1}{2 \cos^2 x} + \frac{2 \sin x \cos x}{2 \cos^2 x} \right)dx\]
\[ = \int e^{2x} \left( \frac{\sec^2 x}{2} + \tan x \right)dx\]
\[\text{ Let e}^{2x} \tan x = t\]
\[ \Rightarrow \left( e^{2x} \sec^2 x + 2 e^{2x} \tan x \right)dx = dt\]
\[ = \left[ \frac{e^{2x} \sec^2 x}{2} + e^{2x} \tan x \right]dx = \frac{dt}{2}\]
\[ \therefore I = \int \frac{dt}{2}\]
\[ = \frac{t}{2} + C\]
\[ = \frac{e^{2x} \tan x}{2} + C\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
Write the primitive or anti-derivative of
\[f\left( x \right) = \sqrt{x} + \frac{1}{\sqrt{x}} .\]
\[\int \left( 2x - 3 \right)^5 + \sqrt{3x + 2} \text{dx} \]
`∫ cos ^4 2x dx `
\[\int\frac{1}{\sqrt{1 + \cos x}} dx\]
` ∫ tan 2x tan 3x tan 5x dx `
\[\int\left( \frac{x + 1}{x} \right) \left( x + \log x \right)^2 dx\]
\[\int\sqrt {e^x- 1} \text{dx}\]
\[\int\frac{x^2 + 3x + 1}{\left( x + 1 \right)^2} dx\]
` ∫ tan^5 x sec ^4 x dx `
\[\int\frac{1}{1 + x - x^2} \text{ dx }\]
\[\int\frac{\sec^2 x}{1 - \tan^2 x} dx\]
` ∫ { x^2 dx}/{x^6 - a^6} dx `
\[\int\frac{x}{x^4 - x^2 + 1} dx\]
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} dx\]
\[\int\frac{x}{\sqrt{x^2 + 6x + 10}} \text{ dx }\]
\[\int\frac{x + 1}{\sqrt{4 + 5x - x^2}} \text{ dx }\]
\[\int\frac{1}{1 + 3 \sin^2 x} \text{ dx }\]
\[\int\frac{1}{3 + 2 \sin x + \cos x} \text{ dx }\]
\[\int\frac{1}{1 - \cot x} dx\]
\[\int\frac{5 \cos x + 6}{2 \cos x + \sin x + 3} \text{ dx }\]
\[\int\frac{1}{3 + 4 \cot x} dx\]
\[\int\frac{8 \cot x + 1}{3 \cot x + 2} \text{ dx }\]
\[\int x \cos^2 x\ dx\]
`int"x"^"n"."log" "x" "dx"`
\[\int2 x^3 e^{x^2} dx\]
\[\int e^\sqrt{x} \text{ dx }\]
\[\int\frac{x^2 \tan^{- 1} x}{1 + x^2} \text{ dx }\]
\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]
\[\int\frac{x^2 + 1}{x^2 - 1} dx\]
\[\int\frac{2x - 3}{\left( x^2 - 1 \right) \left( 2x + 3 \right)} dx\]
\[\int\frac{1}{x \log x \left( 2 + \log x \right)} dx\]
\[\int\frac{x}{\left( x^2 - a^2 \right) \left( x^2 - b^2 \right)} dx\]
\[\int\frac{1}{\left( \sin^{- 1} x \right) \sqrt{1 - x^2}} \text{ dx} \]
\[\int\frac{\sin x}{\sqrt{1 + \sin x}} dx\]
\[\int\frac{1}{\sec x + cosec x}\text{ dx }\]
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} \text{ dx}\]
