integral from x to 1 of cos(sqrt(t))

解答

\int_{x}^{1} cos (t) d t

2 (sin (1) + cos (1) - x sin (x) - cos (x))

求解步骤

\int_{x}^{1} cos (t) d t

使用换元积分法

= \int_{x}^{1} cos (u) \cdot 2 u d u

提出常数:

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

= 2 \cdot \int_{x}^{1} cos (u) u d u

使用分布积分法

= 2 [u sin (u) - \int sin (u) d u]_{x}^{1}

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

= 2 [u sin (u) - (- cos (u))]_{x}^{1}

化简

= 2 [u sin (u) + cos (u)]_{x}^{1}

带入上下限计算后得:

sin (1) + cos (1) - x sin (x) - cos (x)

= 2 (sin (1) + cos (1) - x sin (x) - cos (x))

\int_{- 1}^{0} \frac{t}{( 3 t + 4 ) ^{3}} d t

\int_{0}^{1} - cos (x) d x

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

\int_{0}^{2} 4^{2 x} d x

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