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受欢迎的微积分 >

integral of (7e^{2t})/(1+e^{4t)}

解答

\int \frac{7 e ^{2 t}}{1 + e ^{4 t}} d t

解答

\frac{7}{2} arctan (e^{2 t}) + C

求解步骤

\int \frac{7 e ^{2 t}}{1 + e ^{4 t}} d t

提出常数:

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

= 7 \cdot \int \frac{e ^{2 t}}{1 + e ^{4 t}} d t

使用换元积分法

= 7 \cdot \int \frac{1}{2 ( 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}{2} \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}{2} arctan (u)

u = e^{2 t}

代回

= 7 \cdot \frac{1}{2} arctan (e^{2 t})

化简

7 \cdot \frac{1}{2} arctan (e^{2 t}) : \frac{7}{2} arctan (e^{2 t})

= \frac{7}{2} arctan (e^{2 t})

解答补常数

= \frac{7}{2} arctan (e^{2 t}) + C

作图

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