integral of (cos(sqrt(x)))

解答

\int (cos (x)) d x

2 (x sin (x) + cos (x)) + C

求解步骤

\int cos (x) d x

使用换元积分法

= \int 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 cos (u) u d u

使用分布积分法

= 2 (u sin (u) - \int sin (u) d u)

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

= 2 (u sin (u) - (- cos (u)))

u = x

代回

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

化简

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

解答补常数

= 2 (x sin (x) + cos (x)) + C

x \to \frac{1}{4} lim (sin (\frac{π}{x}))

\frac{d}{d x} ((7 x + 1)^{2})

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

\int x y z^{2} d x

\int sec (x + 1) tan (x + 1) d x