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受欢迎的微积分 >

integral of sqrt(2)asqrt(1+cos(θ))

解答

\int 2 a 1 + cos (θ) d θ

解答

4 a tan (\frac{θ}{2}) cos (\frac{θ}{2}) + C

求解步骤

\int 2 a 1 + cos (θ) d θ

提出常数:

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

= 2 a \cdot \int 1 + cos (θ) d θ

使用换元积分法

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

提出常数:

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

= 2 a \cdot 22 \cdot \int \frac{1}{( 1 + u ^{2} ) 1 + u ^{2}} d u

化简

(1 + u^{2}) 1 + u^{2} : (1 + u^{2})^{\frac{3}{2}}

= 2 a \cdot 22 \cdot \int \frac{1}{( 1 + u ^{2} ) ^{\frac{3}{2}}} d u

使用三角换元法

= 2 a \cdot 22 \cdot \int cos (v) d v

使用积分常见定则:

\int cos (v) d v = sin (v)

= 2 a \cdot 22 sin (v)

代回

= 2 a \cdot 22 sin (arctan (tan (\frac{θ}{2})))

化简

2 a \cdot 22 sin (arctan (tan (\frac{θ}{2}))) : 4 a tan (\frac{θ}{2}) cos (\frac{θ}{2})

= 4 a tan (\frac{θ}{2}) cos (\frac{θ}{2})

解答补常数

= 4 a tan (\frac{θ}{2}) cos (\frac{θ}{2}) + C

作图

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