Problem 58 - Single Variable Calculus

Question

find the given integral. \(\int \tanh x d x\)

Step-by-Step Solution

Answer

The short answer to the given integral is \(\int \tanh x \,d x = \ln|\cosh x| + C\).

Step 1: Rewrite the expression in terms of sinh and cosh functions

Rewrite the given integral in terms of hyperbolic sine and cosine functions: \[ \int \tanh x \,d x = \int\frac{\sinh x}{\cosh x} \,d x . \]

Step 2: Use substitution

Let's perform a substitution. Let \(u = \cosh x\). Then, \(du = \sinh x \,d x\). With this substitution, the integral becomes: \[ \int\frac{\sinh x}{\cosh x} \,d x = \int\frac{1}{u} \,d u. \]

Step 3: Integrate the new expression

Now, integrate the simplified expression with respect to \(u\): \[ \int\frac{1}{u} \,d u = \ln|u| + C. \]

Step 4: Replace u by the original function

Now that we have the antiderivative in terms of \(u\), we substitute back the original function to get our answer in terms of \(x\). We know that \(u = \cosh x\), so the final result is: \[ \ln|\cosh x| + C. \] Therefore, the antiderivative of the given integral is \(\ln|\cosh x| + C\).