Question

To determine the value of the integral, .\(\int {{{\sinh }^2}} x\cosh xdx\)

Short answer

The value of the integral, \(\int {{{\sinh }^2}} x\cosh xdx\) is \(\frac{{{{\sinh }^3}x}}{3} + C\).

Step-by-step solution

Given data

The given integral is, \(\int {{{\sinh }^2}} x\cosh xdx\).

Concept used of Substitution Rule

The Substitution Rule:

If\(u = g(x)\)is a differentiable function whose range is an interval\(l\)and\(f\)is continuous on\(l\), then\(\int f (g(x)){g^\prime }(x)dx = \int f (u)du{.^\prime }\)

Apply substitution rule and simplify

Obtain the value of the integral, \(\int {{{\sinh }^2}} x\cosh xdx\) as follows.

Apply the Substitution Rule to evaluate the integral \(\int {{{\sinh }^2}} x\cosh xdx\).

Use the substitution \(u = \sinh x\). After the differentiation of \(u\), we obtain, \(du = \cosh xdx\).

Apply the above substitutions and the Substitution Rule in the integral as follows.

\(\begin{aligned}{c}\int {{{\sinh }^2}} x\cosh xdx &= \int {{u^2}} du\\ &= \frac{{{u^3}}}{3} + C\\ &= \frac{{{{\sinh }^3}x}}{3} + C\end{aligned}\)

Thus, the value of the integral, \(\int {{{\sinh }^2}} x\cosh xdx\) is \(\frac{{{{\sinh }^3}x}}{3} + C\)..