What is the integral of xtan(5x^2) ?

The integral of xtan(5x^2) is -1/10 ln|cos(5x^2)|+C

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integral of xtan(5x^2)

\int x tan (5 x^{2}) d x

- \frac{1}{10} ln cos (5 x^{2}) + C

Solution steps

\int x tan (5 x^{2}) d x

Apply u-substitution

: \int \frac{tan ( u )}{10} d u

= \int \frac{tan ( u )}{10} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int tan (u) d u

Rewrite using trig identities

= \frac{1}{10} \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

: \int - \frac{1}{v} d v

= \frac{1}{10} \cdot \int - \frac{1}{v} d v

Take the constant out:

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

= \frac{1}{10} (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{10} (- ln ∣ v ∣)

Substitute back

= \frac{1}{10} (- ln cos (5 x^{2}))

Simplify

= - \frac{1}{10} ln cos (5 x^{2})

Add a constant to the solution

= - \frac{1}{10} ln cos (5 x^{2}) + C

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What is the integral of xtan(5x^2) ?
The integral of xtan(5x^2) is -1/10 ln|cos(5x^2)|+C