What is the integral from 0 to pi/3 of 4tan(x) ?

The integral from 0 to pi/3 of 4tan(x) is 4ln(2)

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integral from 0 to pi/3 of 4tan(x)

\int_{0}^{\frac{π}{3}} 4 tan (x) d x

4 ln (2)

Decimal Notation

2.77258 \dots

Solution steps

\int_{0}^{\frac{π}{3}} 4 tan (x) d x

Take the constant out:

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

= 4 \cdot \int_{0}^{\frac{π}{3}} tan (x) d x

Rewrite using trig identities

= 4 \cdot \int_{0}^{\frac{π}{3}} \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

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

= 4 \cdot \int_{1}^{\frac{1}{2}} - \frac{1}{u} d u

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

= 4 (- \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

= 4 (- (- \int_{\frac{1}{2}}^{1} \frac{1}{u} d u))

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 (- (- [ln ∣ u ∣]_{\frac{1}{2}}^{1}))

Simplify

= 4 [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Compute the boundaries:

ln (2)

= 4 ln (2)

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What is the integral from 0 to pi/3 of 4tan(x) ?
The integral from 0 to pi/3 of 4tan(x) is 4ln(2)