Advertisements
Advertisements
Question
\[\int2 x^3 e^{x^2} dx\]
Sum
Advertisements
Solution
\[\int2 x^3 \cdot e^{x^2} dx\]
\[ = \int x^2 \cdot \left( e^{x^2} \right) \cdot \text{ 2x dx }\]
` \text{ Let } x^2" = t `
\[ \Rightarrow \text{ 2x dx } = dt\]
\[ = \int t_I \cdot {e_{II}}^t dt\]
\[ = t \cdot e^t - \int1 \cdot e^t dt\]
\[ = \text{ t e}^t - e^t + C\]
\[ = \text{ x}^2 \text{ e}^{x^2} - e^{x^2} + C\]
\[ = e^{x^2} \left( x^2 - 1 \right) + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\frac{1 + \cos x}{1 - \cos x} dx\]
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
\[\int\frac{1 - \cot x}{1 + \cot x} dx\]
\[\int\frac{2 \cos 2x + \sec^2 x}{\sin 2x + \tan x - 5} dx\]
\[\int\sqrt{1 + e^x} . e^x dx\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} dx\]
\[\int \cot^n {cosec}^2 \text{ x dx } , n \neq - 1\]
` = ∫1/{sin^3 x cos^ 2x} dx`
\[\int\frac{1}{\sin x \cos^3 x} dx\]
\[\int\frac{x^2}{x^6 + a^6} dx\]
\[\int\frac{1}{x \left( x^6 + 1 \right)} dx\]
\[\int\frac{\sin x}{\sqrt{4 \cos^2 x - 1}} dx\]
\[\int\frac{\left( 3 \sin x - 2 \right) \cos x}{5 - \cos^2 x - 4 \sin x} dx\]
\[\int\frac{x^2 \left( x^4 + 4 \right)}{x^2 + 4} \text{ dx }\]
\[\int\frac{1}{1 - \cot x} dx\]
`int"x"^"n"."log" "x" "dx"`
\[\int \cos^3 \sqrt{x}\ dx\]
\[\int\frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x/2} \text{ dx }\]
\[\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }\]
\[\int\sqrt{3 - x^2} \text{ dx}\]
\[\int\frac{x^2 + 1}{\left( x - 2 \right)^2 \left( x + 3 \right)} dx\]
\[\int\frac{x^2}{\left( x - 1 \right) \left( x + 1 \right)^2} dx\]
\[\int\frac{5 x^2 + 20x + 6}{x^3 + 2 x^2 + x} dx\]
\[\int\frac{x}{\left( x + 1 \right) \left( x^2 + 1 \right)} dx\]
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx\]
\[\int\frac{1}{x \left( x^4 - 1 \right)} dx\]
Evaluate the following integral:
\[\int\frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)}dx\]
\[\int\frac{x + 1}{x \left( 1 + x e^x \right)} dx\]
` \int \text{ x} \text{ sec x}^2 \text{ dx is equal to }`
\[\int\frac{\left( 2^x + 3^x \right)^2}{6^x} \text{ dx }\]
\[\int\frac{\sin x}{1 + \sin x} \text{ dx }\]
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
\[\int \sin^4 2x\ dx\]
\[\int x\sqrt{2x + 3} \text{ dx }\]
\[\int\frac{1}{a + b \tan x} \text{ dx }\]
\[\int \log_{10} x\ dx\]
\[\int\frac{1}{x \sqrt{1 + x^n}} \text{ dx}\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }\]
