integral of x/7 sqrt(49+x^2)

解答

\int \frac{x}{7} 49 + x^{2} d x

\frac{1}{21} (49 + x^{2})^{\frac{3}{2}} + C

求解步骤

\int \frac{x}{7} 49 + x^{2} d x

提出常数:

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

= \frac{1}{7} \cdot \int x 49 + x^{2} d x

使用换元积分法

= \frac{1}{7} \cdot \int u^{2} d u

使用幂法则

= \frac{1}{7} \cdot \frac{u ^{3}}{3}

u = 49 + x^{2}

代回

= \frac{1}{7} \cdot \frac{( 49 + x ^{2} ) ^{3}}{3}

化简

\frac{1}{7} \cdot \frac{( 49 + x ^{2} ) ^{3}}{3} : \frac{1}{21} (49 + x^{2})^{\frac{3}{2}}

= \frac{1}{21} (49 + x^{2})^{\frac{3}{2}}

解答补常数

= \frac{1}{21} (49 + x^{2})^{\frac{3}{2}} + C

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