How do you integrate int 1/sqrt(-e^(2x) +9)dx using trigonometric substitution?

1 Answer
Mar 14, 2016

int(1/sqrt(-e^(2x)+9))=(ln(sqrt(9-e^(2x))-3)-x)/3

Explanation:

Here is the corresponding right triangle from which you can extract all the parts for substitution:

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You need to find the following and then systematically replace all x terms (rectangular) with theta terms (trig):

x

dx/(d theta)

sqrt(9-e^(2x))

I'll start you off in the right direction:

Since

sin(theta) =e^x/3

We have then that:

x = ln(3*sin(theta))

Now differentiate with respect to theta and split up the differentials.

You can then rewrite that entire integrand in terms of theta and solve using trigonometric substitution.

I'll give you one more piece and then I gotta sleep!

What is cos(theta)?

Well, referencing the triangle you can see that:

cos (theta) = sqrt(9-e^(2x))/3

This says that:

sqrt(9-e^(2x)) =3*cos(theta)

You'll still need to find d theta

So now you can systematically replace all rectangular terms with
trigonometric terms.

That's your job!