What is the integral of tan(7x) ?

The integral of tan(7x) is -1/7 ln|cos(7x)|+C

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integral of tan(7x)

\int tan (7 x) d x

- \frac{1}{7} ln ∣ cos (7 x) ∣ + C

Solution steps

\int tan (7 x) d x

Rewrite using trig identities

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

Apply u-substitution

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

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{7} ln ∣ u ∣

Substitute back

u = cos (7 x)

= - \frac{1}{7} ln ∣ cos (7 x) ∣

Add a constant to the solution

= - \frac{1}{7} ln ∣ cos (7 x) ∣ + C

\int \frac{2}{x ^{3}} d x

\int 2 x cos (5 x) d x

\frac{\partial}{\partial x} (2 x^{4} y)

\frac{d y}{d x} = - a (y - b)