Integration two different answers

Question

Comments

This is how I got both answers. I just need an explanation showing how the final answers are the same.

Answer 1

(1)

The Function is .

The Hyperbolic cosine function is

Multiply and divide by e^x

Re-write the expression

Now consider u = e^x

Apply derivative on each side.

du = e^x dx

The above integral is in the form of .

Again Substitute u = e^x

Therefore .

Answer 2

(2)

The Function is .

Multiply and divide by cosh(x)

We know hyperbolic sine and cosine identities cosh²(x) - sinh²(x) = 1 then cosh²(x) = 1 + sinh²(x)

Now consider u=sinh(x)

Apply derivative on each side.

du = cosh(x) dx

The above integral is in the form of .

Again Substitute u=sinh(x)

Therefore .

Comments

Please exlpain:

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In the above step,

, we cannot cancel the constant C directly because they may not be same.

This is the actual relationship, .

To prove this, let us consider tan(A - B).

Let us consider, tanA = e^x, tanB = e^-x

Therefore .

Check: take x = 3,