Substitution:

The u substitution:

ex.

ò 2x(x² + 3)² dx

Substitute u = x² + 3 and take its derivative with respect to x, du = 2x dx.

So the integral becomes;

ò u² du = u³/3 + C

Once the solution has been found in terms of u, substitute back into terms of x, therefore the final solution is:

ò 2x(x² + 3)² dx = (x² + 3)³/3 + C

ex.

ò x(x² + 3)² dx

Substitute:

u = x² + 3
du = 2x dx
x dx = 1/2 du

The integral becomes:

The final solution is:

x(x² + 3)² dx =

ex.

Substitute:

u2 = x - 1

u2+ 1 = x

2u du = dx

The integral becomes:

ò (u² + 11) u · 2u du
ò (u² + 1) 2u² du
ò 2u⁴ + 2u² du

The final solution is: