What is the integral of 4tan(7x) ?

The integral of 4tan(7x) is -4/7 ln|cos(7x)|+C

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integral of 4tan(7x)

\int 4 tan (7 x) d x

- \frac{4}{7} ln ∣ cos (7 x) ∣ + C

Solution steps

\int 4 tan (7 x) d x

Take the constant out:

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

= 4 \cdot \int tan (7 x) d x

Rewrite using trig identities

= 4 \cdot \int \frac{sin ( 7 x )}{cos ( 7 x )} d x

Apply u-substitution

: \int - \frac{1}{7 u} d u

= 4 \cdot \int - \frac{1}{7 u} d u

Take the constant out:

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

= 4 (- \frac{1}{7} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 (- \frac{1}{7} ln ∣ u ∣)

Substitute back

u = cos (7 x)

= 4 (- \frac{1}{7} ln ∣ cos (7 x) ∣)

Simplify

4 (- \frac{1}{7} ln ∣ cos (7 x) ∣) : - \frac{4}{7} ln ∣ cos (7 x) ∣

= - \frac{4}{7} ln ∣ cos (7 x) ∣

Add a constant to the solution

= - \frac{4}{7} ln ∣ cos (7 x) ∣ + C

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What is the integral of 4tan(7x) ?
The integral of 4tan(7x) is -4/7 ln|cos(7x)|+C