What is the integral of-3xtan(5x^2) ?

The integral of-3xtan(5x^2) is 3/10 ln|cos(5x^2)|+C

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integral of-3xtan(5x^2)

\int - 3 x tan (5 x^{2}) d x

\frac{3}{10} ln cos (5 x^{2}) + C

Solution steps

\int - 3 x tan (5 x^{2}) d x

Take the constant out:

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

= - 3 \cdot \int x tan (5 x^{2}) d x

Apply u-substitution

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

= - 3 \cdot \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

= - 3 \cdot \frac{1}{10} \cdot \int tan (u) d u

Rewrite using trig identities

= - 3 \cdot \frac{1}{10} \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

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

= - 3 \cdot \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

= - 3 \cdot \frac{1}{10} (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - 3 \cdot \frac{1}{10} (- ln ∣ v ∣)

Substitute back

= - 3 \cdot \frac{1}{10} (- ln cos (5 x^{2}))

Simplify

- 3 \cdot \frac{1}{10} (- ln cos (5 x^{2})) : \frac{3}{10} ln cos (5 x^{2})

= \frac{3}{10} ln cos (5 x^{2})

Add a constant to the solution

= \frac{3}{10} ln cos (5 x^{2}) + C

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What is the integral of-3xtan(5x^2) ?
The integral of-3xtan(5x^2) is 3/10 ln|cos(5x^2)|+C