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受欢迎的微积分 >

integral of-2/(sqrt(81+25t^2))

解答

\int - \frac{2}{81 + 25 t ^{2}} d t

解答

- \frac{2}{5} ln (\frac{5 t + 81 + 25 t ^{2}}{9}) + C

求解步骤

\int - \frac{2}{81 + 25 t ^{2}} d t

提出常数:

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

= - 2 \cdot \int \frac{1}{81 + 25 t ^{2}} d t

使用三角换元法

= - 2 \cdot \int \frac{sec ( u )}{5} d u

提出常数:

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

= - 2 \cdot \frac{1}{5} \cdot \int sec (u) d u

使用积分常见定则:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= - 2 \cdot \frac{1}{5} ln ∣ tan (u) + sec (u) ∣

u = arctan (\frac{5}{9} t)

代回

= - 2 \cdot \frac{1}{5} ln tan (arctan (\frac{5}{9} t)) + sec (arctan (\frac{5}{9} t))

化简

- 2 \cdot \frac{1}{5} ln tan (arctan (\frac{5}{9} t)) + sec (arctan (\frac{5}{9} t)) : - \frac{2}{5} ln (\frac{5 t + 81 + 25 t ^{2}}{9})

= - \frac{2}{5} ln (\frac{5 t + 81 + 25 t ^{2}}{9})

解答补常数

= - \frac{2}{5} ln (\frac{5 t + 81 + 25 t ^{2}}{9}) + C

作图

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