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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用三角恒等式改写

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

提出常数:

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

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

使用换元积分法

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

使用换元积分法

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

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}{v ^{2} + 1} d v

使用积分常见定则:

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

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

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of (sin^2(z))/2-sin(z)

\int \frac{sin ^{2} ( z )}{2} - sin (z) d z

integral of (3x-1)ln(x)

\int (3 x - 1) ln (x) d x

integral of x/(\sqrt[3]{2x+1)}

\int \frac{x}{3 2 x + 1} d x

integral of sqrt(36x^2+36x^4)

\int 36 x^{2} + 36 x^{4} d x

integral of (2-cot(x))(1-csc(x))

\int (2 - cot (x)) (1 - csc (x)) d x