Evaluate the definite integrals- -01 xe x 2 dx

Let I=∫_0^1xe^x^2dx Put x^2 =t ⇒ 2x =dt/dx⇒ dx =dt/2x Limits when x=0 ⇒ t =0 and when x =1⇒ t =1 ∴ I =∫_0^1 xe^t dt/2x=1/2∫_0^1 e^t dt =1/2[e^t]^1_0 =1/2[e^1 e ...

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Standard XII

Mathematics

Functions

Evaluate the ...

Question

Evaluate the definite integrals.
$\int_{0}^{1} x e^{x^{2}} d x .$

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Solution

Let $I = \int_{0}^{1} x e^{x^{2}} d x$
Put $x^{2} = t \Rightarrow 2 x = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{2 x}$
Limits when $x = 0 \Rightarrow t = 0 and when x =1 \Rightarrow t = 1$
$∴ I = \int_{0}^{1} x e^{t} \frac{d t}{2 x} = \frac{1}{2} \int_{0}^{1} e^{t} d t = \frac{1}{2} [e^{t}]_{0}^{1} = \frac{1}{2} [e^{1} - e^{0}] = \frac{1}{2} [e - 1]$

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Evaluate the definite integrals. ∫10xex2dx.

Let I=∫10xex2dx Put x2=t⇒2x=dtdx⇒dx=dt2x Limits when x=0⇒t=0 and when x =1⇒t=1 ∴I=∫10xetdt2x=12∫10etdt=12[et]10=12[e1−e0]=12[e−1]

Evaluate the definite integrals.
$\int_{0}^{1} x e^{x^{2}} d x .$