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受欢迎的微积分 >

integral of (3sec^2(t))/(6+3tan(t))

解答

\int \frac{3 sec ^{2} ( t )}{6 + 3 tan ( t )} d t

解答

ln ∣ 6 + 3 tan (t) ∣ + C

求解步骤

\int \frac{3 sec ^{2} ( t )}{6 + 3 tan ( t )} d t

提出常数:

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

= 3 \cdot \int \frac{sec ^{2} ( t )}{6 + 3 tan ( t )} d t

使用换元积分法

= 3 \cdot \int \frac{1}{3 u} d u

提出常数:

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

= 3 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{3} ln ∣ u ∣

u = 6 + 3 tan (t)

代回

= 3 \cdot \frac{1}{3} ln ∣ 6 + 3 tan (t) ∣

化简

3 \cdot \frac{1}{3} ln ∣ 6 + 3 tan (t) ∣ : ln ∣ 6 + 3 tan (t) ∣

= ln ∣ 6 + 3 tan (t) ∣

解答补常数

= ln ∣ 6 + 3 tan (t) ∣ + C

作图

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