Advertisements
Advertisements
Question
\[\int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x e^x \right)} dx\]
Sum
Advertisements
Solution
\[\int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x \cdot e^x \right)} dx\]
\[\text{Let x e}^x = t\]
\[ \Rightarrow \left( 1 \cdot e^x + \text{x e}^x \right) = \frac{dt}{dx}\]
\[ \Rightarrow \left( x + 1 \right) e^x dx = dt\]
\[Now, \int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x \cdot e^x \right)} dx\]
\[ = \int\frac{dt}{\cos^2 t}\]
\[ = \int \sec^2 \text{t dt}\]
\[ = \tan \left( t \right) + C\]
` = tan ( x e^x) + C `
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]
\[\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx\]
\[\int\frac{1}{\sqrt{1 + \cos x}} dx\]
\[\int\frac{\cos 4x - \cos 2x}{\sin 4x - \sin 2x} dx\]
\[\int x^3 \sin x^4 dx\]
\[\ \int\ x \left( 1 - x \right)^{23} dx\]
\[\int \cos^7 x \text{ dx } \]
\[\int\frac{1}{\sqrt{a^2 + b^2 x^2}} dx\]
\[\int\frac{3 x^5}{1 + x^{12}} dx\]
\[\int\frac{1}{\sqrt{7 - 6x - x^2}} dx\]
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} dx\]
\[\int\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
\[\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} \text{ dx }\]
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
\[\int x^2 \cos 2x\ \text{ dx }\]
\[\int {cosec}^3 x\ dx\]
\[\int x \sin x \cos 2x\ dx\]
\[\int e^x \left( \cos x - \sin x \right) dx\]
\[\int\frac{\sqrt{16 + \left( \log x \right)^2}}{x} \text{ dx}\]
\[\int\left( x + 1 \right) \sqrt{2 x^2 + 3} \text{ dx}\]
\[\int\left( 4x + 1 \right) \sqrt{x^2 - x - 2} \text{ dx }\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
\[\int\frac{x^2 + 1}{\left( x - 2 \right)^2 \left( x + 3 \right)} dx\]
\[\int\frac{dx}{\left( x^2 + 1 \right) \left( x^2 + 4 \right)}\]
\[\int\frac{3}{\left( 1 - x \right) \left( 1 + x^2 \right)} dx\]
\[\int\frac{1}{\cos x \left( 5 - 4 \sin x \right)} dx\]
\[\int e^x \left\{ f\left( x \right) + f'\left( x \right) \right\} dx =\]
\[\int\frac{\sin x}{\cos 2x} \text{ dx }\]
\[\int \cot^4 x\ dx\]
\[\int\frac{1}{\sqrt{x^2 + a^2}} \text{ dx }\]
\[\int\frac{1}{3 x^2 + 13x - 10} \text{ dx }\]
\[\int\frac{1}{\sqrt{3 - 2x - x^2}} \text{ dx}\]
\[\int {cosec}^4 2x\ dx\]
\[\int\frac{\sin^6 x}{\cos x} \text{ dx }\]
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx}\]
\[\int \left( \sin^{- 1} x \right)^3 dx\]
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} \text{ dx}\]
