four basic/easy integrals

[golfman44]

(x^4+3)^4x^3dx

let u = x^4 + 3
du = 4x^3 dx

1/4 du = x^3 dx

1/4 * integral (u)^4 du

1/4 * integral 1/5(u)^5 + c

answer: 1/20 (x^4 + 3)^5 + c


THE NEXT THREE I AM STRUGGLING WITH... HELP IS APPRECIATED!

2. integral (x^2 - 2)^2 dx

3. integral sin^5(2x) cos(2x) dx

4. integral [ (9x^6 + 5x^4 + 7) / (x^3) ] dx

 

[galactus]

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

Expand out and integrate term by term.

$\displaystyle \int(x^{2}-2)^{2}dx=\int x^{4}dx -4\int x^{2}dx+4\int dx$

3. $\displaystyle \int sin^{5}(2x) cos(2x) dx$

This is a basic substitutions. Okey-doke.

Let $\displaystyle u=sin(2x), \;\ \frac{du}{2}=cos(2x)dx$

Make the subs and we get:

$\displaystyle \frac{1}{2}\int u^{5}du$

Now, it's easier?.

4. $\displaystyle \int \frac{9x^{6} + 5x^{4} + 7}{x^{3}} dx$

Long divide and then integrate term by term.

$\displaystyle 9\int x^{3}dx+5\int xdx+7\int x^{-3}dx$