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受欢迎的微积分 >

integral of (16)/(9+25x^2)

解答

\int \frac{16}{9 + 25 x ^{2}} d x

解答

\frac{16}{15} arctan (\frac{5}{3} x) + C

求解步骤

\int \frac{16}{9 + 25 x ^{2}} d x

提出常数:

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

= 16 \cdot \int \frac{1}{9 + 25 x ^{2}} d x

使用换元积分法

= 16 \cdot \int \frac{1}{15 ( u ^{2} + 1 )} d u

提出常数:

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

= 16 \cdot \frac{1}{15} \cdot \int \frac{1}{u ^{2} + 1} d u

使用积分常见定则:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= 16 \cdot \frac{1}{15} arctan (u)

u = \frac{5}{3} x

代回

= 16 \cdot \frac{1}{15} arctan (\frac{5}{3} x)

化简

16 \cdot \frac{1}{15} arctan (\frac{5}{3} x) : \frac{16}{15} arctan (\frac{5}{3} x)

= \frac{16}{15} arctan (\frac{5}{3} x)

解答补常数

= \frac{16}{15} arctan (\frac{5}{3} x) + C

作图

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