Find the definite integral of the following using a suitable substitution:
1) (x^2)/sqroot(x^3-1)dx
2) xe^(x^2)dx
3) (ln(x))^(7/2)/(x)dx
Those would be indefinite integrals since your are not evaluating for any values.
My tip for substitution is always look at what part you can take the derivative of and it will replace something else in the function.
Ill show an example with the first.
$\displaystyle \int \frac{x^2}{\sqrt{x^3-1}} dx$
Notice we can take the following substitution
$\displaystyle u = x^3-1$
Taking the derivative of this yields.
$\displaystyle du = 3x^2 dx$
Now notice in the numerator there is an $\displaystyle x^2$ to replace it properly we simply get rid of the 3 in the derivative to make it fit properly
$\displaystyle \frac{1}{3}du = x^2 dx$
Now the numerator will be replaced properly when we do the substitution
$\displaystyle \frac{1}{3}\int \frac{1}{\sqrt{u}} du$
Now integrating this yields.
$\displaystyle \frac{2}{3} \cdot \sqrt{u}$
Now we substitute back into the answer
$\displaystyle \frac{2\sqrt{x^3-1}}{3}$