What is the integral from pi to 2pi of sin^3(x) ?

The integral from pi to 2pi of sin^3(x) is -4/3

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integral from pi to 2pi of sin^3(x)

\int_{π}^{2 π} sin^{3} (x) d x

- \frac{4}{3}

Decimal Notation

- 1.33333 \dots

Solution steps

\int_{π}^{2 π} sin^{3} (x) d x

Rewrite using trig identities

= \int_{π}^{2 π} (1 - cos^{2} (x)) sin (x) d x

Apply u-substitution

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int_{- 1}^{1} u^{2} d u = \frac{2}{3}

= - 2 + \frac{2}{3}

Simplify

- 2 + \frac{2}{3} : - \frac{4}{3}

= - \frac{4}{3}

\int_{0}^{2 π} \frac{1}{2 + cos ( θ )} d θ

\int_{1}^{e} (\frac{x ^{2} + 4}{x}) d x

\int_{- 1}^{1} x e^{x^{4}} d x

\int_{3}^{5} (\frac{( ln ( x ) )}{x})^{4} d x

\int_{- 125}^{1} \frac{1}{x ^{\frac{1}{3}}} d x

What is the integral from pi to 2pi of sin^3(x) ?
The integral from pi to 2pi of sin^3(x) is -4/3