Evaluate the integral using any method:

(Integral)sec^3x/tanx dx
I started it out and got secx(1tan^2x)/tanx. I know I just have to continue simplifying and finding the integral, but I'm stuck on the next couple of steps.

Also, I have another question witht he same directions:(integral)sin^3/cosx dx. What would I do here?

Thank you!

I did:

sec^3x/tanx
= (1/cos^3 x)(cosx/sinx)
= 1/((sinx)(cos^2 x)

so I went to my favourite integrator page and got

http://integrals.wolfram.com/index.jsp?expr=1%2F%28%28sin%28x%29cos%5E2%28x%29%29&random=false

You should realize that when Wolfram says log(...)
they really mean ln(...)

for your 2nd
http://integrals.wolfram.com/index.jsp?expr=sin%5E3%28x%29%2Fcos%28x%29&random=false

thanks for the first part. i wanted something that was more step-by-step for the second one.

sin^3/cosx

= (sinx/cos)(sin^2x)
= tanx sin^2 x

then
http://integrals.wolfram.com/index.jsp?expr=%28tan%28x%29%28sin%28x%29%5E2%29&random=false
same as above

To evaluate the integral ∫sec^3(x)/tan(x) dx, you have made a good start by simplifying the integrand to sec(x)(tan^2(x))/tan(x). Now, let's continue the simplification and find the integral.

First, simplify the expression to sec(x)tan(x). Note that sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x). So, sec(x)tan(x) becomes (1/cos(x))(sin(x)/cos(x)).

Next, multiply the fractions together to get sin(x)/cos^2(x).

Now you have the expression sin(x)/cos^2(x) as the integrand. To evaluate this integral, you can use a substitution.

Let u = cos(x), then du = -sin(x) dx. Rearrange this equation to get sin(x) dx = -du.

Now substitute these values into the integral. The integral becomes:

∫ (sin(x)/cos^2(x)) dx = ∫ (-du/u^2) = -∫ du/u^2

Integrating -1/u^2 gives -(1/u) + C, where C is the constant of integration.

Substituting back u = cos(x), we get:

-(1/cos(x)) + C.

So the final answer to the integral is -(1/cos(x)) + C.

Now, let's move to the other question:

To evaluate the integral ∫sin^3(x)/cos(x) dx, use the following trigonometric identity: sin^2(x) = 1 - cos^2(x).

We can rewrite the integral as ∫sin(x)sin^2(x)/cos(x) dx.

Now substitute sin^2(x) = 1 - cos^2(x), yielding:

∫sin(x)(1 - cos^2(x))/cos(x) dx.

Distribute sin(x) into the parentheses:

∫sin(x) - sin(x)cos^2(x)/cos(x) dx.

Simplify sin(x)cos^2(x)/cos(x) to sin(x)cos(x):

∫sin(x) - sin(x)cos(x) dx.

Now, you can separate the integral into two part:

∫sin(x) dx - ∫sin(x)cos(x) dx.

The first integral is straightforward:

∫sin(x) dx = -cos(x) + C1 (constant of integration).

The second integral can be evaluated using a substitution.

Let u = cos(x), then du = -sin(x) dx. Rearrange this equation to get -sin(x) dx = du.

Now substitute these values into the integral:

∫sin(x)cos(x) dx = ∫ -du = -u + C2 (constant of integration).

Substituting u = cos(x), we get:

- cos(x) + C2.

Hence, the final answer to the integral is -cos(x) + C1 - cos(x) + C2, which can be simplified to -2cos(x) + C, where C is the constant of integration.