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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

\int \frac{1}{sin ( x ) + 2 cos ( x ) + 3} 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-9cos^2(x)

\int - 9 cos^{2} (x) d x

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\int \frac{- e ^{- l n (x)}}{x} d x

integral of ((x-9))/(x^2-x-2)

\int \frac{( x - 9 )}{x ^{2} - x - 2} d x

integral of sqrt(8)

\int 8 d x

integral of (18x^2)/(\sqrt[3]{x^3+4)}

\int \frac{18 x ^{2}}{3 x ^{3} + 4} d x