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受欢迎的微积分 >

integral of 60(1+x^2)(7+15x+5x^3)^{16}

解答

\int 60 (1 + x^{2}) (7 + 15 x + 5 x^{3})^{16} d x

解答

\frac{4}{17} (5 x^{3} + 15 x + 7)^{17} + C

求解步骤

\int 60 (1 + x^{2}) (7 + 15 x + 5 x^{3})^{16} d x

提出常数:

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

= 60 \cdot \int (1 + x^{2}) (7 + 15 x + 5 x^{3})^{16} d x

使用换元积分法

= 60 \cdot \int \frac{u ^{16}}{15} d u

提出常数:

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

= 60 \cdot \frac{1}{15} \cdot \int u^{16} d u

使用幂法则

= 60 \cdot \frac{1}{15} \cdot \frac{u ^{17}}{17}

u = 7 + 15 x + 5 x^{3}

代回

= 60 \cdot \frac{1}{15} \cdot \frac{( 7 + 15 x + 5 x ^{3} ) ^{17}}{17}

化简

60 \cdot \frac{1}{15} \cdot \frac{( 7 + 15 x + 5 x ^{3} ) ^{17}}{17} : \frac{4}{17} (5 x^{3} + 15 x + 7)^{17}

= \frac{4}{17} (5 x^{3} + 15 x + 7)^{17}

解答补常数

= \frac{4}{17} (5 x^{3} + 15 x + 7)^{17} + C

作图

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