1.

$\displaystyle \int (x^{2} + 4)^{5} 2xdx$

Do the anti-derivative on the integrand.

$\displaystyle \frac{(x^{2} + 4)^{6}}{6} + C$ Final Answer

and this problem

2.

$\displaystyle \int (x^{3} - 3x)^{1/2}(x^{2} - 1)dx$

Next taking the derivative of the highest power in the integrand $\displaystyle x^{3}$ (BUT WAIT - 3 is not the derivative)

$\displaystyle (x^{3} - 3x)^{1/2} 3(x^{2} - 1)dx$

Next flipping the $\displaystyle 3$ so it becomes $\displaystyle \frac{1}{3}$

$\displaystyle \frac{1}{3} \int (x^{3} - 3x)^{1/2} 3(x^{2} - 1)dx$

The stuff to the right of the left linear factor (integrand) disappears. Now we do the anti-derivative on the integrand.

$\displaystyle \frac{1}{3} [\frac{(x^{3} - 3x)^{3/2}}{\frac{3}{2}} + C]$

$\displaystyle \frac{2}{9}(x^{3} - 3x)^{3/2} + C$ Final Answer

As you can see that in the 2nd problem the $\displaystyle \frac{1}{3}$ is a flipped version of 3 which came from the $\displaystyle x^{3}$ in the beginning (since that was the largest power).

But in the 1st problem the highest power $\displaystyle x^{3}$ is not made into 3 and flipped.

But I see a problem in the reasoning: $\displaystyle x^{3}$ differentiated is [tex]3x^{2}[tex] not 3. So why does the 3 come about? Where did it come from?