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  • Evaluate the definite integrals in Exercises 1 to 20. Integral 0 to 1 x e^x^2 dx

Evaluate the definite integrals in Exercises 1 to 20.

    Q15.    \int_0^1xe^{x^2}dx

Answers (1)

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Given integral: \int_0^1xe^{x^2}dx

Consider the integral \int xe^{x^2}dx

Putting x^2 = t which gives, 2xdx =dt

As, x\rightarrow0 ,t \rightarrow0  and  as x\rightarrow1 ,t \rightarrow1.

So, we have now:

\therefore I = \frac{1}{2}\int_0^1 e^t dt 

= \frac{1}{2}\int e^t dt = \frac{1}{2} e^t

So, we have the function of x, f(x) = \frac{1}{2} e^t

Now, by Second fundamental theorem of calculus, we have

I = f(1) -f(0)

= \frac{1}{2}e^1 -\frac{1}{2}e^0 = \frac{1}{2}(e-1)

Posted by

Divya Prakash Singh

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