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受欢迎的微积分 >

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

解答

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

解答

arctan (\frac{1}{2} (tan (\frac{x}{2}) + 1)) + C

求解步骤

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

使用换元积分法

= \int \frac{2}{u ^{2} + 2 u + 5} d u

提出常数:

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

= 2 \cdot \int \frac{1}{u ^{2} + 2 u + 5} d u

对

u^{2} + 2 u + 5

配方:

(u + 1)^{2} + 4

= 2 \cdot \int \frac{1}{( u + 1 ) ^{2} + 4} d u

使用换元积分法

= 2 \cdot \int \frac{1}{v ^{2} + 4} d v

使用换元积分法

= 2 \cdot \int \frac{1}{2 ( w ^{2} + 1 )} d w

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{w ^{2} + 1} d w

使用积分常见定则:

\int \frac{1}{w ^{2} + 1} d w = arctan (w)

= 2 \cdot \frac{1}{2} arctan (w)

代回

= 2 \cdot \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} ) + 1}{2})

化简

2 \cdot \frac{1}{2} arctan (\frac{tan ( \frac{x}{2} ) + 1}{2}) : arctan (\frac{1}{2} (tan (\frac{x}{2}) + 1))

= arctan (\frac{1}{2} (tan (\frac{x}{2}) + 1))

解答补常数

= arctan (\frac{1}{2} (tan (\frac{x}{2}) + 1)) + C

作图

流行的例子

integral of sin(x)+cos(2x)

\int sin (x) + cos (2 x) d x

integral of (e^{9x})/(e^{9x)+7}

\int \frac{e ^{9 x}}{e ^{9 x} + 7} d x

integral of 5e^x+4/x

\int 5 e^{x} + \frac{4}{x} d x

integral of (e^{9x})

\int (e^{9 x}) d x

integral of x^{3/4}-1/(x^{3/4)}

\int x^{\frac{3}{4}} - \frac{1}{x ^{\frac{3}{4}}} d x