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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

= \int u cos (u^{2} - 1) d u

因式分解

u^{2} - 1 : - (- u^{2} + 1)

= \int u cos (- (- u^{2} + 1)) d u

化简

u cos (- (- u^{2} + 1)) : u cos (1 - u^{2})

= \int u cos (1 - u^{2}) d u

使用换元积分法

= \int - \frac{cos ( v )}{2} d v

提出常数:

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

= - \frac{1}{2} \cdot \int cos (v) d v

使用积分常见定则:

\int cos (v) d v = sin (v)

= - \frac{1}{2} sin (v)

代回

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

化简

- \frac{1}{2} sin (1 - (x + 1)^{2}) : \frac{1}{2} sin (x^{2} + 2 x)

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

解答补常数

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

作图

流行的例子

integral of (8x^3)/(sqrt(4x^2+9))

\int \frac{8 x ^{3}}{4 x ^{2} + 9} d x

integral of tan(x+1)

\int tan (x + 1) d x

integral of cos(x)sqrt(sin(x)+1)

\int cos (x) sin (x) + 1 d x

integral of (e^{1/(x^2)}+2x+ln(x))/(x^3)

\int \frac{e ^{\frac{1}{x ^{2}}} + 2 x + ln ( x )}{x ^{3}} d x

integral of 3/((1-2x)^2)

\int \frac{3}{( 1 - 2 x ) ^{2}} d x