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受欢迎的微积分 >

integral of 3/(2cos(x)+1)

解答

\int \frac{3}{2 cos ( x ) + 1} d x

解答

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

求解步骤

\int \frac{3}{2 cos ( x ) + 1} 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}{2 cos ( x ) + 1} d x

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

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

代回

= 3 \cdot 2 \cdot \frac{1}{3} \frac{ln \frac{t a n ( \frac{x}{2} )}{3} + 1}{2} - \frac{ln \frac{t a n ( \frac{x}{2} )}{3} - 1}{2}

化简

3 \cdot 2 \cdot \frac{1}{3} \frac{ln \frac{t a n ( \frac{x}{2} )}{3} + 1}{2} - \frac{ln \frac{t a n ( \frac{x}{2} )}{3} - 1}{2} : 3 ln \frac{1}{3} tan (\frac{x}{2}) + 1 - 3 ln \frac{1}{3} tan (\frac{x}{2}) - 1

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

解答补常数

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

作图

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