Evaluate the following definite Integral: ∫(xe^{2x} + sin(πx)/2)dx, x ∈ [0, 1 ]

Question

Evaluate the following definite Integral:

\(\int\limits_0^1\Big(\text xe^{2\text x}+sin\cfrac{\pi \text x}2\Big) \)dx

Answer 1

First split the integral

Now integrate by parts the first one

For this we have to apply integration by parts

Let u and v be two functions then

∫udv = uv - ∫vdu

To choose the first function u we use “ILATE” rule

That is

I=inverse trigonometric function

L=logarithmic function

A=algebraic function

T=trigonometric functions

E=exponential function

So in this preference, the first function is chosen to make the integration simpler.

Now, In the given question x² is an algebraic function and it is chosen as u(A comes first in “ILATE” rule)

So first let us integrate the equation and then let us substitute the limits in it.

Remember 

∫ sinx = -cos x and ∫ cos x = sin x