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受欢迎的微积分 >

integral of (21x^2)/(4(x^3+5)^{5/4)}

解答

\int \frac{21 x ^{2}}{4 ( x ^{3} + 5 ) ^{\frac{5}{4}}} d x

解答

- \frac{7}{( x ^{3} + 5 ) ^{\frac{1}{4}}} + C

求解步骤

\int \frac{21 x ^{2}}{4 ( x ^{3} + 5 ) ^{\frac{5}{4}}} d x

提出常数:

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

= \frac{21}{4} \cdot \int \frac{x ^{2}}{( x ^{3} + 5 ) ^{\frac{5}{4}}} d x

使用换元积分法

= \frac{21}{4} \cdot \int \frac{1}{3 u ^{\frac{5}{4}}} d u

提出常数:

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

= \frac{21}{4} \cdot \frac{1}{3} \cdot \int \frac{1}{u ^{\frac{5}{4}}} d u

使用幂法则

= \frac{21}{4} \cdot \frac{1}{3} (- \frac{4}{u ^{\frac{1}{4}}})

u = x^{3} + 5

代回

= \frac{21}{4} \cdot \frac{1}{3} (- \frac{4}{( x ^{3} + 5 ) ^{\frac{1}{4}}})

化简

\frac{21}{4} \cdot \frac{1}{3} (- \frac{4}{( x ^{3} + 5 ) ^{\frac{1}{4}}}) : - \frac{7}{( x ^{3} + 5 ) ^{\frac{1}{4}}}

= - \frac{7}{( x ^{3} + 5 ) ^{\frac{1}{4}}}

解答补常数

= - \frac{7}{( x ^{3} + 5 ) ^{\frac{1}{4}}} + C

作图

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