integral of x^2cos(1/2 x)

解答

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

2 x^{2} sin (\frac{x}{2}) - 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2})) + C

求解步骤

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

化简

\frac{1}{2} x : \frac{x}{2}

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

使用分布积分法

= 2 x^{2} sin (\frac{x}{2}) - \int 4 x sin (\frac{x}{2}) d x

\int 4 x sin (\frac{x}{2}) d x = 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}))

= 2 x^{2} sin (\frac{x}{2}) - 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}))

解答补常数

= 2 x^{2} sin (\frac{x}{2}) - 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2})) + C

\int_{0}^{1} \frac{6}{1 + t ^{2}} d t

\frac{d}{d x} (\frac{cos ( x )}{x} + \frac{x}{cos ( x )})

d er i v a t i v e \frac{1}{4 x ^{2} + 1}

x \to 0 + lim (\frac{x ^{2}}{9} - \frac{2}{x})

\int \frac{1}{x ^{\frac{1}{2}} + x ^{\frac{3}{2}}} d x