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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.9
Evaluate the definite integrals in Exercises 176 to 195.
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Question : 190 of 271
Marks: +1, -0
∫01xex2dx\int\limits_{0}^{1}xe^{x^{2}}dx
Solution:  
Let I=∫01xex2dxI=\int\limits_{0}^{1}xe^{x^{2}}dx =12∫012xex2dx=\frac{1}{2}\int\limits_{0}^{1}2xe^{x^{2}}dx
Put x2=t⇒2x dx=dtx^{2}=t\Rightarrow 2x\,dx=dt
When x=0, t=0x=0,\ t=0 ; When x=1, t=1x=1,\ t=1
∴I=12∫01etdt\therefore I=\frac{1}{2}\int\limits_{0}^{1}e^{t}dt =[12et]01=\left[\frac{1}{2}e^{t}\right]_{0}^{1}
=12[e1−e0]=\frac{1}{2}[e^{1}-e^{0}] =12[e−1]=\frac{1}{2}[e-1]
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