Solution assignment 02 Integration by substitution

Solve:

$\displaystyle\int{5(2x+10)^3}dx$

First we rewrite the integral:

$\displaystyle\int{5(2x+10)^3}dx=5\displaystyle\int{(2x+10)^3}dx$

and use the transform:

$u=2x+10$

from which it follows:

$\displaystyle\frac{du}{dx}=2$

or:

$dx=\displaystyle\frac{1}{2}du$

These results can be substituted in the original integral and we get:

$5\displaystyle\int{u^3.\displaystyle\frac{1}{2}}du=\displaystyle\frac{5}{2}.\displaystyle\frac{1}{4}u^4+C$

After 'transforming back' we get:

$\displaystyle\frac{5}{8}(2x+10)^4+C$

We can verify this result by differentiation. The chain rule has to be applied.