What is the integral of tan(6x+1) ?

The integral of tan(6x+1) is -1/6 ln|cos(6x+1)|+C

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integral of tan(6x+1)

\int tan (6 x + 1) d x

- \frac{1}{6} ln ∣ cos (6 x + 1) ∣ + C

Solution steps

\int tan (6 x + 1) d x

Apply u-substitution

= \int tan (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int tan (u) d u

Rewrite using trig identities

= \frac{1}{6} \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

= \frac{1}{6} \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{1}{6} (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{6} (- ln ∣ v ∣)

Substitute back

= \frac{1}{6} (- ln ∣ cos (6 x + 1) ∣)

Simplify

= - \frac{1}{6} ln ∣ cos (6 x + 1) ∣

Add a constant to the solution

= - \frac{1}{6} ln ∣ cos (6 x + 1) ∣ + C

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d er i v a t i v e arcsin (\frac{x}{7})

d er i v a t i v e y = 4^{- x} sin (x)

\int \frac{9 - x ^{2}}{7 x ^{3} + x} d x

x \to 2 lim (\frac{x ^{7} - 128}{x - 2})

What is the integral of tan(6x+1) ?
The integral of tan(6x+1) is -1/6 ln|cos(6x+1)|+C