What is the integral of xsech^2(x^2) ?

The integral of xsech^2(x^2) is 1/2 tanh(x^2)+C

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integral of xsech^2(x^2)

\int x sech^{2} (x^{2}) d x

\frac{1}{2} tanh (x^{2}) + C

Solution steps

\int x sech^{2} (x^{2}) d x

Apply u-substitution

: \int \frac{sech ^{2} ( u )}{2} d u

= \int \frac{sech ^{2} ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sech^{2} (u) d u

Use the common integral:

\int sech^{2} (u) d u = tanh (u)

= \frac{1}{2} tanh (u)

Substitute back

u = x^{2}

= \frac{1}{2} tanh (x^{2})

Add a constant to the solution

= \frac{1}{2} tanh (x^{2}) + C

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What is the integral of xsech^2(x^2) ?
The integral of xsech^2(x^2) is 1/2 tanh(x^2)+C