Solve ∫ operatorname{sech}1/4xtanh1/4x wrt x

$$\int \operatorname { sech } \frac { 1 } { 4 } x \tanh \frac { 1 } { 4 } x d x$$

Evaluate

$$\frac{\operatorname{sech}(\frac{1}{4})\tanh(\frac{1}{4})x^{3}}{3}+С$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int \operatorname{sech}(\frac{1}{4})x^{2}\tanh(\frac{1}{4})\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\operatorname{sech}(\frac{1}{4})\tanh(\frac{1}{4})\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$\operatorname{sech}(\frac{1}{4})\tanh(\frac{1}{4})\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{\operatorname{sech}(\frac{1}{4})\tanh(\frac{1}{4})x^{3}}{3}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{\operatorname{sech}(\frac{1}{4})\tanh(\frac{1}{4})x^{3}}{3}+С$$

Differentiate w.r.t. x

$$\operatorname{sech}(\frac{1}{4})\tanh(\frac{1}{4})x^{2}$$