The correct option is C 12( x^2 1)e^x^2 + c C) #160; ∫ x ^ 3 e ^ x ^ 2 d x put x ^ 2 = t ⇒ 2 x d x = d t 1 2 ∫ t e ^ t d t = ...

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Integration by Parts

∫x3ex2dx =

Question

$\int x^{3} e^{x^{2}} d x =$

\frac{1}{2} (x^{2} + 1) e^{x^{2}} + c

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(x^{2} + 1) e^{x^{2}} + c

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\frac{1}{2} (x^{2} - 1) e^{x^{2}} + c

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(x^{2} - 1) e^{x^{2}} + c

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Solution

The correct option is C $\frac{1}{2} (x^{2} - 1) e^{x^{2}} + c$
C) $\int x^{3} e^{x^{2}} d x$

put $x^{2} = t \Rightarrow 2 x d x = d t$

$\frac{1}{2} \int t e^{t} d t = \frac{1}{2} [(t e^{t}) - \int e^{t} d t]$ {integration by parts}

$= \frac{1}{2} [t e^{t} - e^{t}] + c$

$= \frac{1}{2} (x^{2} e^{x^{2}} - e^{x^{2}}) + c$

$= \frac{1}{2} (x^{2} - 1) e^{x^{2}} + c$

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Question :

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I = \int \frac{x^{3}}{\sqrt{x^{2} + 2 x + 5}} d x = \frac{1}{6} f (x) - \frac{A}{120} g (x) + C

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