Advertisements
Advertisements
Question
\[\int e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx\]
Sum
Solution
\[\text{ Let I }= \int e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx\]
\[\text{ Also let e}^x \times \frac{1}{x^2} = t \]
\[\text{ Diff both sides w . r . t x }\]
\[ e^x \times \frac{1}{x^2} + e^x \left( \frac{- 2}{x^3} \right) = \frac{dt}{dx}\]
\[ \Rightarrow e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx = dt\]
\[ \therefore \int e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx = \int dt\]
\[ = t + C\]
\[ = \frac{e^x}{x^2} + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\sqrt{x}\left( x^3 - \frac{2}{x} \right) dx\]
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
\[\int\frac{\left( x + 1 \right)\left( x - 2 \right)}{\sqrt{x}} dx\]
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
\[\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx\]
\[\int\frac{1}{1 + \sqrt{x}} dx\]
\[\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx\]
\[\int \cot^5 \text{ x } {cosec}^4 x\text{ dx }\]
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 - 1}} dx\]
\[\int\frac{e^x}{e^{2x} + 5 e^x + 6} dx\]
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} dx\]
\[\int\frac{x^2 + x + 1}{x^2 - x} dx\]
\[\int\frac{x^3 + x^2 + 2x + 1}{x^2 - x + 1}\text{ dx }\]
\[\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} \text{ dx }\]
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
\[\int\frac{1}{3 + 2 \sin x + \cos x} \text{ dx }\]
\[\int\frac{1}{4 + 3 \tan x} dx\]
\[\int x e^x \text{ dx }\]
\[\int\frac{\log \left( \log x \right)}{x} dx\]
`int"x"^"n"."log" "x" "dx"`
\[\int\frac{\log x}{x^n}\text{ dx }\]
\[\int \log_{10} x\ dx\]
\[\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} dx\]
\[\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }\]
\[\int\frac{2x + 1}{\left( x + 1 \right) \left( x - 2 \right)} dx\]
\[\int\frac{5x}{\left( x + 1 \right) \left( x^2 - 4 \right)} dx\]
\[\int\frac{x + 1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} dx\]
If \[\int\frac{\cos 8x + 1}{\tan 2x - \cot 2x} dx\]
If \[\int\frac{1}{\left( x + 2 \right)\left( x^2 + 1 \right)}dx = a\log\left| 1 + x^2 \right| + b \tan^{- 1} x + \frac{1}{5}\log\left| x + 2 \right| + C,\] then
\[\int\frac{1}{\left( \sin^{- 1} x \right) \sqrt{1 - x^2}} \text{ dx} \]
\[\int\sqrt{\text{ cosec x} - 1} \text{ dx }\]
\[\int\sqrt{\frac{1 + x}{x}} \text{ dx }\]
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
\[\int\frac{\cos x}{\frac{1}{4} - \cos^2 x} \text{ dx }\]
\[\int\sqrt{1 + 2x - 3 x^2}\text{ dx } \]
\[\int\frac{x^2}{\sqrt{1 - x}} \text{ dx }\]
