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

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

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integral of 7tan(7x)

\int 7 tan (7 x) d x

- ln ∣ cos (7 x) ∣ + C

Solution steps

\int 7 tan (7 x) d x

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

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

= 7 \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 7 (- \frac{1}{7} ln ∣ u ∣)

Substitute back

u = cos (7 x)

= 7 (- \frac{1}{7} ln ∣ cos (7 x) ∣)

Simplify

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

= - ln ∣ cos (7 x) ∣

Add a constant to the solution

= - ln ∣ cos (7 x) ∣ + C

\int 3 tan (x) d x

\frac{d}{d t} (\frac{t}{t ^{2} + 16})

- x^{3} + x^{2} + 16 x, 4 x

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

\frac{d}{d x} (- 3 x^{2} - 5 x + 1)