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受欢迎的微积分 >

integral of 7/(3+7x^2)

解答

\int \frac{7}{3 + 7 x ^{2}} d x

解答

\frac{7}{3} arctan (\frac{7}{3} x) + C

求解步骤

\int \frac{7}{3 + 7 x ^{2}} d x

提出常数:

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

= 7 \cdot \int \frac{1}{3 + 7 x ^{2}} d x

使用换元积分法

= 7 \cdot \int \frac{1}{21 ( 1 + u ^{2} )} d u

提出常数:

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

= 7 \cdot \frac{1}{21} \cdot \int \frac{1}{1 + u ^{2}} d u

使用积分常见定则:

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

= 7 \cdot \frac{1}{21} arctan (u)

u = \frac{7}{3} x

代回

= 7 \cdot \frac{1}{21} arctan (\frac{7}{3} x)

化简

7 \cdot \frac{1}{21} arctan (\frac{7}{3} x) : \frac{7}{3} arctan (\frac{7}{3} x)

= \frac{7}{3} arctan (\frac{7}{3} x)

解答补常数

= \frac{7}{3} arctan (\frac{7}{3} x) + C

作图

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(1 + x) \frac{d y}{d x} - x y = x - x^{2}