Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)2 Marks
Question
Evalute : $\int x^3 e^{x^2} d x$
✓
Answer
Let $I=\int x^3 e^{x^2} d x=\int x^2 e^{x^2} \cdot x d x$ Put $x^2=t \quad \therefore 2 x d x=d t$ $ \begin{aligned} & \therefore x d x=\frac{d t}{2}\\ & \therefore I=\int t e^t \cdot \frac{d t}{2}=\frac{1}{2} \int t e^t d t \\ & =\frac{1}{2}\left[t \int e^t d t-\int\left\{\frac{d}{d t}(t) \int e^t d t\right\} d t\right] \\ & =\frac{1}{2}\left[t e^t-\int 1 \cdot e^t d t\right] \\ & =\frac{1}{2}\left[t e^t-e^t\right]+c \\ & =\frac{1}{2}(t-1) e^t+c \\ & =\frac{1}{2}\left(x^2-1\right) e^{x^2}+c . \\ \end{aligned} $
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