integral from 0 to pi/2 of sin^5(xco)s^{12}x

解答

\int_{0}^{\frac{π}{2}} sin^{5} (x co) s^{12} x

\frac{s ^{12} ( 18 sin ^{5} ( \frac{π co}{2} ) + 40 sin ^{3} ( \frac{π co}{2} ) + 240 sin ( \frac{π co}{2} ) + 150 π co cos ^{3} ( \frac{π co}{2} ) - 225 π co cos ( \frac{π co}{2} ) - 45 π co cos ^{5} ( \frac{π co}{2} ) )}{450 c ^{2} o ^{2}}

求解步骤

\int_{0}^{\frac{π}{2}} sin^{5} (x co) s^{12} x d x

提出常数:

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

= s^{12} \cdot \int_{0}^{\frac{π}{2}} sin^{5} (co x) x d x

使用换元积分法

= s^{12} \cdot \int_{0}^{\frac{π co}{2}} \frac{u sin ^{5} ( u )}{c ^{2} o ^{2}} d u

提出常数:

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

= s^{12} \frac{1}{c ^{2} o ^{2}} \cdot \int_{0}^{\frac{π co}{2}} u sin^{5} (u) d u

使用分布积分法

= s^{12} \frac{1}{c ^{2} o ^{2}} [u (- cos (u) + \frac{2}{3} cos^{3} (u) - \frac{1}{5} cos^{5} (u)) - \int \frac{1}{15} (- 15 cos (u) + 10 cos^{3} (u) - 3 cos^{5} (u)) d u]_{0}^{\frac{π co}{2}}

\int \frac{1}{15} (- 15 cos (u) + 10 cos^{3} (u) - 3 cos^{5} (u)) d u = \frac{1}{225} (- 9 sin^{5} (u) - 120 sin (u) - 20 sin^{3} (u))

= s^{12} \frac{1}{c ^{2} o ^{2}} [u (- cos (u) + \frac{2}{3} cos^{3} (u) - \frac{1}{5} cos^{5} (u)) - \frac{1}{225} (- 9 sin^{5} (u) - 120 sin (u) - 20 sin^{3} (u))]_{0}^{\frac{π co}{2}}

带入上下限计算后得:

\frac{1}{25} sin^{5} (\frac{π co}{2}) + \frac{4}{45} sin^{3} (\frac{π co}{2}) + \frac{8}{15} sin (\frac{π co}{2}) + \frac{π co cos ^{3} ( \frac{π co}{2} )}{3} - \frac{π co}{2} cos (\frac{π co}{2}) - \frac{π co}{10} cos^{5} (\frac{π co}{2})

= s^{12} \frac{1}{c ^{2} o ^{2}} (\frac{1}{25} sin^{5} (\frac{π co}{2}) + \frac{4}{45} sin^{3} (\frac{π co}{2}) + \frac{8}{15} sin (\frac{π co}{2}) + \frac{π co cos ^{3} ( \frac{π co}{2} )}{3} - \frac{π co}{2} cos (\frac{π co}{2}) - \frac{π co}{10} cos^{5} (\frac{π co}{2}))

化简

= \frac{s ^{12} ( 18 sin ^{5} ( \frac{π co}{2} ) + 40 sin ^{3} ( \frac{π co}{2} ) + 240 sin ( \frac{π co}{2} ) + 150 π co cos ^{3} ( \frac{π co}{2} ) - 225 π co cos ( \frac{π co}{2} ) - 45 π co cos ^{5} ( \frac{π co}{2} ) )}{450 c ^{2} o ^{2}}

\frac{\partial}{\partial x} (x^{\frac{1}{2}} \cdot y^{\frac{1}{2}})

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

\frac{\partial}{\partial x} (z^{x^{y}})

t an g e n t f (x) = \frac{x}{x ^{2} + 81}

a re a x = 2 y^{2} - 5, x = y^{2} + 4