Evaluating Indifinite Integral

[PolyFX]

Homework Statement

$$\int \sec^{2}(3x)e^{\tan(3x)}dx$$

Homework Equations

The method I am trying to use is integration by substitution(u substitution).

The Attempt at a Solution

I start out by making u = tan(3x)

So i end up having $$\int \sec^{2}(3x)e^{u}dx$$

Stuck after here though. How do I simplify this further?


The final solution as per my professors solution sheet should be $$1/3e^{tan(3x)} + C$$

However, I cannot seem to figure out why this is the solution.


-Thank you

 

[Unto]

change sec^{2} 3x into 1 + tan^{2}

Then make substitutions

 

[Dick]

If u=tan(3x), then what's du?

 

[Mark44]

change sec^{2} 3x into 1 + tan^{2}

Then make substitutions

This is not a helpful suggestion.

 

[PolyFX]

Would du be $$sec^{2}(3x) (3dx)$$?

 

[Dick]

Yes, it would. Can you use that to finish? Solve that for dx and put it into the original integral.

 

[PolyFX]

hi, sorry for the late reply.


Solving for dx I got

dx= du/(Sec^2(3x)) x 3

So

Sec^2(3x)e^u du/sec^2(3x) x 3


sec^2(3x) cancel each other out?

so I'm left with

e^u(du)(1/3) = 1/3e^u(du)

so 1/3e^(tan(3x))+C?

Is my approach correct?

-Thank You

 

[Dick]

hi, sorry for the late reply.


Solving for dx I got

dx= du/(Sec^2(3x)) x 3

So

Sec^2(3x)e^u du/sec^2(3x) x 3


sec^2(3x) cancel each other out?

so I'm left with

e^u(du)(1/3) = 1/3e^u(du)

so 1/3e^(tan(3x))+C?

Is my approach correct?

-Thank You

Yes, yes, yes.