Need some help on a couple integrations

[DaAzNJRiCh]

You dont need to do them all but one would be nice. I want to see how to do it rather than just get answers. Thanks.

Integral of [(e^x + cos(x))^2] dx
I tried to expand it and then got confused with integration by parts but I think you use that

Integral of e^(2x) cos(x) dx
u substitution???

Integral of sin^2 (7x) dx
I heard something about the half angle formula but isnt that like sin (x/2)?

Integral of cos^4 (x) sin^3 (x) dx
Should i change the cos^4 (x) to like (1-sin(x))^2 or something?

Integral of [(4-x^2)^.5]/ x dx

 

[galactus]

For the 1st one:

Expand it out and then integrate. May make it easier.

\(\displaystyle \L\\(e^{x}+cos(x))^{2}=e^{2x}+2e^{x}+cos^{2}(x)\)


For the 2nd one:

\(\displaystyle \L\\\int{e^{2x}cos(x)dx\)

Use Integration by parts:

Start with:

\(\displaystyle \L\\u=e^{2x}, \;\ dv=cos(x)dx, \;\ du=2e^{2x}dx, \;\ v=sin(x)\)


For #3:

You can use \(\displaystyle \L\\\int{sin^{2}(x)}dx=\frac{1}{2}x-\frac{1}{4}sin(2x)+C\)


For #4:

Try \(\displaystyle \L\\\int{cos^{4}(x)sin^{3}(x)dx}=\int{cos^{4}(x)(1-cos^{2}(x))sin(x)dx\)

Now, let \(\displaystyle u=cos(x), \;\ du={-}sin(x)dx\)

For #5:

\(\displaystyle \L\\\int\frac{\sqrt{4-x^{2}}}{x}dx\)

You can try trig substitution.

Let \(\displaystyle \L\\x=2sin({\theta}), dx=2cos({\theta})d{\theta}\)

Just some thoughts. Have fun.