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

解答

\int_{y}^{2} x^{3} sin (x) d x

4 cos (2) + 6 sin (2) + y^{3} cos (y) - 3 y^{2} sin (y) - 6 y cos (y) + 6 sin (y)

求解步骤

\int_{y}^{2} x^{3} sin (x) d x

使用分布积分法

= [- x^{3} cos (x) - \int - 3 x^{2} cos (x) d x]_{y}^{2}

\int - 3 x^{2} cos (x) d x = - 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))

= [- x^{3} cos (x) - (- 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))]_{y}^{2}

化简

= [- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))]_{y}^{2}

带入上下限计算后得:

4 cos (2) + 6 sin (2) + y^{3} cos (y) - 3 y^{2} sin (y) - 6 y cos (y) + 6 sin (y)

= 4 cos (2) + 6 sin (2) + y^{3} cos (y) - 3 y^{2} sin (y) - 6 y cos (y) + 6 sin (y)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 7 cos (x) d x

\int_{1}^{2} x^{3} + 3 x^{2} - 4 x d x

\int_{0}^{5} 3 t - 8 d t

\int_{1}^{\infty} \frac{x}{1 + x ^{4}} d x

\int_{0}^{7} ∣ x - 6 ∣ d x