What is the integral from pi/4 to pi/3 of tan(x) ?

The integral from pi/4 to pi/3 of tan(x) is 1/2 ln(2)

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integral from pi/4 to pi/3 of tan(x)

\int_{\frac{π}{4}}^{\frac{π}{3}} tan (x) d x

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

Decimal

0.34657 \dots

Solution steps

\int_{\frac{π}{4}}^{\frac{π}{3}} tan (x) d x

Rewrite using trig identities

= \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - (- [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{2}{2}})

Simplify

- (- [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{2}{2}}) : [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{1}{2}}

= [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{1}{2}}

Compute the boundaries:

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

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

What is the integral from pi/4 to pi/3 of tan(x) ?
The integral from pi/4 to pi/3 of tan(x) is 1/2 ln(2)