1. integrals question

Hi i have having problems with a few integral questions. I am just totally lost of what to do.

Let a > 0 and a not equal to 1. Find the indefinite integrals:

________________ + C

________________ + C

and

Evaluate the definite integral.

________________

and

Evaluate the indefinite integral

________________ + C

and

Find the indefinite integral

________________ + C

I know these are alot of problems but I don't know how to solve for any of them.

Thank you

2. Some of those look ugly, but I can certainly help with the last one.

(I'm just going to put INT for integration)

INT ( x sin (2x) ) dx

You can integrate this by parts.

So let u = x and v'= sin 2x
then u' = 1 and v = 0.5 cos 2x

Then INT ( u v' ) dx = uv - INT ( v u' )

=> INT ( x sin (2x) ) dx = 0.5 x cos(2x) - 0.5 INT ( cos (2x) ) dx

= 0.5 x cos(2x) + 0.25 sin 2x + C

3. I can see how to do the second one as well though that will take more working...

INT ( cosx [sinx]^a ) dx

You can integrate this by parts.

So let u = [sinx]^a and v' = cosx
Then u' = a cosx [sinx]^a-1 and v = sinx

Then INT ( u v' ) dx = uv - INT ( u' v )

=> INT ( cosx [sinx]^a ) dx = [sinx]^a+1 - a INT ( cosx [sinx]^a ) dx

Now you can see this last term is just equal to what you started with times a.

=> {a+1} INT ( cosx [sinx]^a ) dx = [sinx]^a+1

=> INT ( cosx [sinx]^a ) dx = [ (sinx)^a+1 ] / [a+1] + C

4. Originally Posted by killasnake
Hi i have having problems with a few integral questions. I am just totally lost of what to do.

Let a > 0 and a not equal to 1. Find the indefinite integrals:

________________ + C
int cos(x) a^(sin(x)) dx = int cos(x) e^[ln(a) sin(x)] dx

Now you should recognise that ln(a) cos(x) is the derivative of ln(a) sin(x),
so:

int cos(x) e^[ln(a) sin(x)] dx = int d/dx{ (1/ln(a) e^[ln(a) sin(x)] } dx

..........................=1/ln(a) e^[ln(a) sin(x)] +C

RonL

5. The methods below work, but I would suggest you do this first:
Note that d/dx(sin(x)) = cos(x).

So for the first two problems I would recommend putting
u = sin(x)
then look at the integrals and see what you can do with them.

For the third problem, a similar approach will work:
u = cos(x)

-Dan