What is the integral of tan(y) ?

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

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integral of tan(y)

\int tan (y) d y

- ln ∣ cos (y) ∣ + C

Solution steps

\int tan (y) d y

Rewrite using trig identities

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

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 (y)

= - ln ∣ cos (y) ∣

Add a constant to the solution

= - ln ∣ cos (y) ∣ + C

\int \frac{x ^{6} e ^{x} - 4 x ^{5}}{x ^{6}} d x

x \to π - lim ((x - π) csc (x))

f (x) = \frac{4 x - 7}{28 x ^{5}}

y = cos^{3} (x)

\int_{0}^{\infty} \frac{1}{x ^{3}} d x