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Evaluate the following problems using second fundamental theorem: $\int\limits_{0}^{1} x^{2} e^{x} dx$

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Toolbox:
  • If $F(x)=\int \limits_a^x f(t)dt $ then $\int \limits_a^b f(x) dx=F(b)-F(a)$
$\int\limits_{0}^{1} x^{2} e^{x} dx$
Step 1:
$\int\limits_{0}^{1} x^{2} e^{x} dx=\int \limits_0^1 udv$
Where $u=x\;and\; dv=e^xdx$
$du=2x \;dx\; and \;v=e^x$
Integrating by parts
$I=uv-\int \limits _0^1 vdu=x^2 e^x \bigg]_0^1-\int \limits _0^1 e^x.2x dx$
Step 2:
$I_1=2 \int \limits _0^1 e^x x dx=2 \int \limits_0^1 udv$
Where $u=x\;and\; dv=e^xdx$
$du=2x \;dx\; and \;v=e^x$
Integrating by parts
$\qquad=2[uv-\int \limits _0^1 vdu]$
$\qquad=2 [(xe^x)_0^1-\int \limits_0^1 e^x dx]$
$\qquad=2[xe^x-e^x]_0^1$
Step 3:
$\therefore I= x^2 e^x\bigg]_0^1-2 \bigg[xe^x-e^x\bigg]_0^1$
$\qquad=e-2[e-e-(-1)]$
$\qquad=e-2$
answered Aug 14, 2013 by meena.p
 

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