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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate : $ \int \limits_0^1 x^4.e^x.dx$

$(a)\;9e+24 \\(b)\;e+24 \\(c)\;9e-24 \\ (d)\;e-24$

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A)
$\int u. v.dx= uv_1- u'v_2+u''v_3-u'''v_4+u''''v_5$
Where $ v_1= \int vdx \qquad u'= [\large\frac{du}{dx}]$
$ v_2= \int v_1dx \qquad u''= [\large\frac{du'}{dx}]$
$u=x^4,v=e^x$
=> $x^4.e^x -4x^3e^x +12x^2e^x -24xe^x+24e^x+c$
$\Rightarrow I = \bigg [ e^x ( x^4 - 4x^3 + 12x^2 - 24 x + 24) \bigg ]_{0}^{1}$
$\quad = 9e - 24$
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