What is the integral of (tan(θ)) ?

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

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integral of (tan(θ))

\int (tan (θ)) d θ

- ln ∣ cos (θ) ∣ + C

Solution steps

\int tan (θ) d θ

Rewrite using trig identities

= \int \frac{sin ( θ )}{cos ( θ )} d θ

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = cos (θ)

= - ln ∣ cos (θ) ∣

Add a constant to the solution

= - ln ∣ cos (θ) ∣ + C

x \to 0 lim (\frac{sin ( π x )}{x})

\int \frac{1}{x ( ln ( x ) ) ^{5}} d x

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) (x + 3 x^{3}))

\int \frac{1}{( x + 3 ) ln ( x + 3 )} d x

e^{a x}