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受欢迎的微积分 >

integral of 3/(sqrt(1+e^{2x))}

解答

\int \frac{3}{1 + e ^{2 x}} d x

解答

- 3 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1) + C

求解步骤

\int \frac{3}{1 + e ^{2 x}} d x

提出常数:

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

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

使用换元积分法

= 3 \cdot \int \frac{1}{u ^{2} - 1} d u

因式分解

u^{2} - 1 : - (- u^{2} + 1)

= 3 \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

提出常数:

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

= 3 (- \int \frac{1}{- u ^{2} + 1} d u)

使用积分常见定则:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= 3 (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

u = 1 + e^{2 x}

代回

= 3 (- (\frac{ln 1 + e ^{2 x} + 1}{2} - \frac{ln 1 + e ^{2 x} - 1}{2}))

化简

3 (- (\frac{ln 1 + e ^{2 x} + 1}{2} - \frac{ln 1 + e ^{2 x} - 1}{2})) : - 3 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1)

= - 3 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1)

解答补常数

= - 3 (\frac{1}{2} ln 1 + e^{2 x} + 1 - \frac{1}{2} ln 1 + e^{2 x} - 1) + C

作图

流行的例子

integral from 0 to 1 of x^7(1+x^8)^7

\int_{0}^{1} x^{7} (1 + x^{8})^{7} d x

integral of (x+9)/(sqrt(1-2x-2x^2))

\int \frac{x + 9}{1 - 2 x - 2 x ^{2}} d x

integral from 0 to 1 of y/(e^{9y)}

\int_{0}^{1} \frac{y}{e ^{9 y}} d y

斜率 y=2+4x^2-2x^3

s l o p e y = 2 + 4 x^{2} - 2 x^{3}

\int - 3 d x