What is the integral from 1/2 to 1/3 of cos(pit) ?

The integral from 1/2 to 1/3 of cos(pit) is -(2-sqrt(3))/(2pi)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 1/2 to 1/3 of cos(pit)

\int_{\frac{1}{2}}^{\frac{1}{3}} cos (π t) d t

- \frac{2 - 3}{2 π}

Decimal

- 0.04264 \dots

Solution steps

\int_{\frac{1}{2}}^{\frac{1}{3}} cos (π t) d t

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

= - \int_{\frac{1}{3}}^{\frac{1}{2}} cos (π t) d t

Apply u-substitution

: \int_{\frac{π}{3}}^{\frac{π}{2}} cos (u) \frac{1}{π} d u

= - \int_{\frac{π}{3}}^{\frac{π}{2}} cos (u) \frac{1}{π} d u

Take the constant out:

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

= - \frac{1}{π} \cdot \int_{\frac{π}{3}}^{\frac{π}{2}} cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= - \frac{1}{π} [sin (u)]_{\frac{π}{3}}^{\frac{π}{2}}

Compute the boundaries:

1 - \frac{3}{2}

= - \frac{1}{π} (1 - \frac{3}{2})

Simplify

= - \frac{2 - 3}{2 π}

\int_{- 1}^{1} 8 - 8 x^{2} d x

\int_{0}^{t} u e^{2 u} sin (3 u) d u

\int_{1}^{6} \frac{1}{x - 4} d x

\int_{1}^{3} (x^{2} - x + 6) d x

\int_{- 1}^{2} \frac{1}{x ^{\frac{2}{3}}} d x

What is the integral from 1/2 to 1/3 of cos(pit) ?
The integral from 1/2 to 1/3 of cos(pit) is -(2-sqrt(3))/(2pi)