Question

Integrate the function w..r. to x : $x^{3}\cdotp \log x$

Answer 1

Solution:

To integrate the given function we must use integration by parts

Formula:

$\int U.V dx=U\int V dx-\int (\frac{dU}{dx}\int V dx)dx$

$\int x^{3}.log\:x dx=log x\int x^{3} dx-\int (\frac{d\:log x}{dx}\int x^{3} dx)dx\\\\=\frac{x^{4}log\:x}{4}-\int\frac{1×x^{4}}{4 x} dx\\\\ =\frac{x^{4}log\:x}{4}-\frac{1}{4} \int x^{3} dx\\\\ =\frac{x^{4}log\:x}{4}-\frac{1}{4}\frac{x^{4}}{4} +C$
So,
$\int x^{3}.log\:x dx=\frac{x^{4}log\:x}{4}-\frac{x^{4}}{16} +C$