What is the integral from-pi/3 to 0 of tan(x) ?

The integral from-pi/3 to 0 of tan(x) is -ln(2)

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integral from-pi/3 to 0 of tan(x)

\int_{- \frac{π}{3}}^{0} tan (x) d x

- ln (2)

Solution steps

\int_{- \frac{π}{3}}^{0} tan (x) d x

Rewrite using trig identities

= \int_{- \frac{π}{3}}^{0} \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

: \int_{\frac{1}{2}}^{1} - \frac{1}{u} d u

= \int_{\frac{1}{2}}^{1} - \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}{2}}^{1} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Compute the boundaries:

ln (2)

= - ln (2)

\int_{1}^{4} x - \frac{1}{x ^{2}} d x

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

\int_{- 1}^{1} (x^{5} - x^{6}) d x

\int_{1}^{3} k x^{2} - 2 d x

\int_{0}^{2} - 2 x + 2 d x

What is the integral from-pi/3 to 0 of tan(x) ?
The integral from-pi/3 to 0 of tan(x) is -ln(2)