What is the integral from 0 to x of tan(x) ?

The integral from 0 to x of tan(x) is -ln|cos(x)|

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integral from 0 to x of tan(x)

\int_{0}^{x} tan (x) d x

- ln ∣ cos (x) ∣

Solution steps

\int_{0}^{x} tan (x) d x

Rewrite using trig identities

= \int_{0}^{x} \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

: \int_{1}^{c o s (x)} - \frac{1}{u} d u

= \int_{1}^{c o s (x)} - \frac{1}{u} d u

Take the constant out:

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

= - \int_{1}^{c o s (x)} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - [ln ∣ u ∣]_{1}^{c o s (x)}

Compute the boundaries:

ln ∣ cos (x) ∣

= - ln ∣ cos (x) ∣

\int_{- 1}^{0} (2 x + e^{x}) d x

\int_{0}^{1} x e^{- 1} d x

\int_{1}^{4} (x^{4} + 5 x^{2}) d x

\int_{0}^{2} x^{3} (2 + x^{4})^{3} d x

\int_{0}^{2} 3 x^{3} - 2 x^{2} - e^{x} d x

What is the integral from 0 to x of tan(x) ?
The integral from 0 to x of tan(x) is -ln|cos(x)|