What is the integral from-pi/4 to 0 of tan(x) ?

The integral from-pi/4 to 0 of tan(x) is -1/2 ln(2)

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integral from-pi/4 to 0 of tan(x)

\int_{- \frac{π}{4}}^{0} tan (x) d x

- \frac{1}{2} ln (2)

Decimal

- 0.34657 \dots

Solution steps

\int_{- \frac{π}{4}}^{0} tan (x) d x

Rewrite using trig identities

= \int_{- \frac{π}{4}}^{0} \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - [ln ∣ u ∣]_{\frac{2}{2}}^{1}

\frac{2}{2} = \frac{1}{2}

= - [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Compute the boundaries:

\frac{1}{2} ln (2)

= - \frac{1}{2} ln (2)

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What is the integral from-pi/4 to 0 of tan(x) ?
The integral from-pi/4 to 0 of tan(x) is -1/2 ln(2)