Integral problem

Question

Solve step by step.

Answer 1

The Function is .

The Hyperbolic sine function is

Substitute the value of sinh(x) in the above function.

Now consider .

Apply derivative on both sides.

Then

Let us consider

Comapre the coefficients of u.

A+B = 0

A = -B

Comapre the coefficients of Constants.

A - B = 2

2A = 2

A = 1 then B = -1.

         (Using the Logarithmic rule )

Substitute .

We know that  .

Therefore .

Comments

I don't understand the part after substituting u=e^x near the end. After substituting you multiply the exponent by 2 and divide it by 2? How is that then the same as tanh(x) which has e^2x in the numerator and demoninator not e^2(x/2)?

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The Standard algebraic expression of Hyperbolic function of tanh(x) is  .

Now coming to our solution, after substituting , we get

Multiply and divide with 2 in the power of the exponent function.

The term resembles the Standard algebraic expression of Hyperbolic function of tanh(x) where we have x/2 instead of x.

Then we can write .

Therefore we can write

Therefore .