What is the integral from 0 to pi of 4(4-3cos(x))^{1/3}sin(x) ?

The integral from 0 to pi of 4(4-3cos(x))^{1/3}sin(x) is 7*7^{1/3}-1

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integral from 0 to pi of 4(4-3cos(x))^{1/3}sin(x)

\int_{0}^{π} 4 (4 - 3 cos (x))^{\frac{1}{3}} sin (x) d x

7 \cdot 7^{\frac{1}{3}} - 1

Decimal

12.39051 \dots

Solution steps

\int_{0}^{π} 4 (4 - 3 cos (x))^{\frac{1}{3}} sin (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}^{π} (4 - 3 cos (x))^{\frac{1}{3}} sin (x) d x

Apply u-substitution

: \int_{1}^{- 1} - (4 - 3 u)^{\frac{1}{3}} d u

= 4 \cdot \int_{1}^{- 1} - (4 - 3 u)^{\frac{1}{3}} d u

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

= 4 (- \int_{- 1}^{1} - (4 - 3 u)^{\frac{1}{3}} d u)

Take the constant out:

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

= 4 (- (- \int_{- 1}^{1} (4 - 3 u)^{\frac{1}{3}} d u))

Apply u-substitution

: \int_{7}^{1} - \frac{v ^{\frac{1}{3}}}{3} d v

= 4 (- (- \int_{7}^{1} - \frac{v ^{\frac{1}{3}}}{3} d v))

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

= 4 (- (- (- \int_{1}^{7} - \frac{v ^{\frac{1}{3}}}{3} d v)))

Take the constant out:

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

= 4 (- (- (- (- \frac{1}{3} \cdot \int_{1}^{7} v^{\frac{1}{3}} d v))))

Apply the Power Rule

: [\frac{3}{4} v^{\frac{4}{3}}]_{1}^{7}

= 4 (- (- (- (- \frac{1}{3} [\frac{3}{4} v^{\frac{4}{3}}]_{1}^{7}))))

Simplify

4 (- (- (- (- \frac{1}{3} [\frac{3}{4} v^{\frac{4}{3}}]_{1}^{7})))) : \frac{4}{3} [\frac{3}{4} v^{\frac{4}{3}}]_{1}^{7}

= \frac{4}{3} [\frac{3}{4} v^{\frac{4}{3}}]_{1}^{7}

Compute the boundaries:

\frac{21 \cdot 7 ^{\frac{1}{3}}}{4} - \frac{3}{4}

= \frac{4}{3} (\frac{21 \cdot 7 ^{\frac{1}{3}}}{4} - \frac{3}{4})

Simplify

= 7 \cdot 7^{\frac{1}{3}} - 1

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\int_{0}^{\frac{5}{12}} \frac{1}{144 x ^{2} + 25} d x

\int_{0}^{1} x 39 x^{2} + 25 d x

\int_{- 1}^{1} \frac{1}{2 π} e^{- \frac{x ^{2}}{2}} d x

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

What is the integral from 0 to pi of 4(4-3cos(x))^{1/3}sin(x) ?
The integral from 0 to pi of 4(4-3cos(x))^{1/3}sin(x) is 7*7^{1/3}-1