integral of x*cos(kx)

解答

\int x \cdot cos (k x) d x

\frac{1}{k ^{2}} (k x sin (k x) + cos (k x)) + C

求解步骤

\int x cos (k x) d x

使用换元积分法

= \int \frac{u cos ( u )}{k ^{2}} d u

提出常数:

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

= \frac{1}{k ^{2}} \cdot \int u cos (u) d u

使用分布积分法

= \frac{1}{k ^{2}} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{k ^{2}} (u sin (u) - (- cos (u)))

u = k x

代回

= \frac{1}{k ^{2}} (k x sin (k x) - (- cos (k x)))

化简

= \frac{1}{k ^{2}} (k x sin (k x) + cos (k x))

解答补常数

= \frac{1}{k ^{2}} (k x sin (k x) + cos (k x)) + C

t an g e n t - 10 x^{4} + x^{3}

d er i v a t i v e y = x^{\frac{13}{14}}

n = 1 \sum \infty ln ((n + 1)^{2})

2 y^{^{''}} + 5 y^{^{'}} - 3 y = 0, y (0) = 1, y^{^{'}} (0) = 4

n = 0 \sum \infty \frac{1}{3 ^{n + 1}}