Advertisements
Advertisements
Question
\[\int 5^{x + \tan^{- 1} x} . \left( \frac{x^2 + 2}{x^2 + 1} \right) dx\]
Sum
Advertisements
Solution
\[\int 5^{x + \tan^{- 1} x} \cdot \left( \frac{x^2 + 2}{x^2 + 1} \right)dx\]
\[\text{Let x} + \tan^{- 1} x = t\]
\[\left( 1 + \frac{1}{1 + x^2} \right) = \frac{dt}{dx}\]
\[ \Rightarrow \left( \frac{x^2 + 1 + 1}{x^2 + 1} \right)dx = dt\]
\[ \Rightarrow \left( \frac{x^2 + 2}{x^2 + 1} \right)dx = dt\]
\[Now, \int 5^{x + \tan^{- 1} x} \cdot \left( \frac{x^2 + 2}{x^2 + 1} \right)dx\]
\[ = \int 5^t dt\]
\[ = \frac{5^t}{\log 5} + C\]
\[ = \frac{5^{x + \tan^1 x}}{\log 5} + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]
\[\int \left( \tan x + \cot x \right)^2 dx\]
\[\int \sin^{- 1} \left( \frac{2 \tan x}{1 + \tan^2 x} \right) dx\]
\[\int\frac{1 + \cos x}{1 - \cos x} dx\]
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
\[\int\frac{x^2 + x + 5}{3x + 2} dx\]
\[\int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx\]
\[\int\text{sin mx }\text{cos nx dx m }\neq n\]
\[\int\frac{\cos x}{\cos \left( x - a \right)} dx\]
\[\int\frac{1}{\sqrt{1 - x^2}\left( 2 + 3 \sin^{- 1} x \right)} dx\]
\[\int\frac{\tan x}{\sqrt{\cos x}} dx\]
\[\int\frac{1}{1 + \sqrt{x}} dx\]
\[\int\frac{\sin \left( \tan^{- 1} x \right)}{1 + x^2} dx\]
\[\int\frac{1}{x^2 \left( x^4 + 1 \right)^{3/4}} dx\]
` ∫ tan^5 x sec ^4 x dx `
\[\int\frac{1}{\sqrt{2x - x^2}} dx\]
\[\int\frac{\cos x - \sin x}{\sqrt{8 - \sin2x}}dx\]
\[\int\frac{x + 1}{x^2 + x + 3} dx\]
\[\int\frac{1}{4 \cos^2 x + 9 \sin^2 x}\text{ dx }\]
\[\int\frac{\log \left( \log x \right)}{x} dx\]
\[\int e^\sqrt{x} \text{ dx }\]
\[\int {cosec}^3 x\ dx\]
\[\int e^x \left( \cos x - \sin x \right) dx\]
∴\[\int e^{2x} \left( - \sin x + 2 \cos x \right) dx\]
\[\int\left( x + 2 \right) \sqrt{x^2 + x + 1} \text{ dx }\]
\[\int\frac{1}{x \log x \left( 2 + \log x \right)} dx\]
\[\int\frac{1}{1 + x + x^2 + x^3} dx\]
\[\int\frac{1}{\left( x - 1 \right) \sqrt{x^2 + 1}} \text{ dx }\]
\[\int\frac{\sin^6 x}{\cos^8 x} dx =\]
\[\int\frac{x^4 + x^2 - 1}{x^2 + 1} \text{ dx}\]
\[\int\sin x \sin 2x \text{ sin 3x dx }\]
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} \text{ dx }\]
\[\int \cot^4 x\ dx\]
\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]
\[\int\frac{1}{\sqrt{x^2 - a^2}} \text{ dx }\]
\[\int\sqrt{x^2 - a^2} \text{ dx}\]
\[\int x\sqrt{1 + x - x^2}\text{ dx }\]
\[\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }\]
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
