Question

Integrate the function w..r. to x : $x^{2} \sin x$

Answer 1

Solution:

This integration can be done by parts,as we know that we must follow the rules of Integration by parts as taking the first function from the rule ILATE

$\intU.Vdx= U\intV\:dx-\int(\frac{dU}{dx}\int V\:dx)dx\\\\\int x^{2} sin\:x dx = x^{2}\int sin\:x\:dx-\int(\frac{dx^{2}}{dx}\int sin\:x dx =- x^{2} cos\:x+\int2x (-cos\:x) dx\\\\x^{2} cos\:x-2\int x (cos\:x) dx ...eq1$

Look at the second term it again look like U.V form,again integration by parts has to be done

$\int x cos x.dx= x\int cos x\:dx-\int(\frac{dx}{dx}\int cos x\:dx)dx\\\\= x\:sin\:x+cos\:x$

put this integration in the eq2

$\intx^{2} \sin x\:dx=- x^{2}cos\:x+2x\:sin\:x+2cos\:x+C$