What is the integral of 5tan(5x) ?

The integral of 5tan(5x) is -ln|cos(5x)|+C

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integral of 5tan(5x)

\int 5 tan (5 x) d x

- ln ∣ cos (5 x) ∣ + C

Solution steps

\int 5 tan (5 x) d x

Take the constant out:

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

= 5 \cdot \int tan (5 x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{1}{5 u} d u

= 5 \cdot \int - \frac{1}{5 u} d u

Take the constant out:

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

= 5 (- \frac{1}{5} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 (- \frac{1}{5} ln ∣ u ∣)

Substitute back

u = cos (5 x)

= 5 (- \frac{1}{5} ln ∣ cos (5 x) ∣)

Simplify

5 (- \frac{1}{5} ln ∣ cos (5 x) ∣) : - ln ∣ cos (5 x) ∣

= - ln ∣ cos (5 x) ∣

Add a constant to the solution

= - ln ∣ cos (5 x) ∣ + C

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