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受欢迎的微积分 >

integral of 4xcos^3(x^2)sin(x^2)

解答

\int 4 x cos^{3} (x^{2}) sin (x^{2}) d x

解答

- \frac{1}{2} cos^{4} (x^{2}) + C

求解步骤

\int 4 x cos^{3} (x^{2}) sin (x^{2}) d x

提出常数:

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

= 4 \cdot \int x cos^{3} (x^{2}) sin (x^{2}) d x

使用换元积分法

= 4 \cdot \int \frac{cos ^{3} ( u ) sin ( u )}{2} d u

提出常数:

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

= 4 \cdot \frac{1}{2} \cdot \int cos^{3} (u) sin (u) d u

使用换元积分法

= 4 \cdot \frac{1}{2} \cdot \int - v^{3} d v

提出常数:

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

= 4 \cdot \frac{1}{2} (- \int v^{3} d v)

使用幂法则

= 4 \cdot \frac{1}{2} (- \frac{v ^{4}}{4})

代回

= 4 \cdot \frac{1}{2} (- \frac{cos ^{4} ( x ^{2} )}{4})

化简

4 \cdot \frac{1}{2} (- \frac{cos ^{4} ( x ^{2} )}{4}) : - \frac{1}{2} cos^{4} (x^{2})

= - \frac{1}{2} cos^{4} (x^{2})

解答补常数

= - \frac{1}{2} cos^{4} (x^{2}) + C

作图

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