What is the integral of (12x+2)tan(15x^2+5x) ?

The integral of (12x+2)tan(15x^2+5x) is -2/5 ln|cos(15x^2+5x)|+C

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integral of (12x+2)tan(15x^2+5x)

\int (12 x + 2) tan (15 x^{2} + 5 x) d x

- \frac{2}{5} ln cos (15 x^{2} + 5 x) + C

Solution steps

\int (12 x + 2) tan (15 x^{2} + 5 x) d x

Apply u-substitution

: \int \frac{2 tan ( u )}{5} d u

= \int \frac{2 tan ( u )}{5} d u

Take the constant out:

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

= \frac{2}{5} \cdot \int tan (u) d u

Rewrite using trig identities

= \frac{2}{5} \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

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

= \frac{2}{5} \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{2}{5} (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{2}{5} (- ln ∣ v ∣)

Substitute back

= \frac{2}{5} (- ln cos (15 x^{2} + 5 x))

Simplify

= - \frac{2}{5} ln cos (15 x^{2} + 5 x)

Add a constant to the solution

= - \frac{2}{5} ln cos (15 x^{2} + 5 x) + C

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What is the integral of (12x+2)tan(15x^2+5x) ?
The integral of (12x+2)tan(15x^2+5x) is -2/5 ln|cos(15x^2+5x)|+C