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受欢迎的微积分 >

integral of sqrt(2+2cos(4x))

解答

\int 2 + 2 cos (4 x) d x

解答

sin (2 x) + C

求解步骤

\int 2 + 2 cos (4 x) d x

2 + 2 cos (4 x) = 2 1 + cos (4 x)

= \int 2 1 + cos (4 x) d x

提出常数:

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

= 2 \cdot \int 1 + cos (4 x) d x

使用换元积分法

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

提出常数:

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

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

使用三角恒等式改写

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

提出常数:

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

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

cos^{2} (u) = (cos (u)),

假定

cos (u) \geq 0

= 2 \frac{1}{2} 2 \cdot \int cos (u) d u

使用积分常见定则:

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

= 2 \frac{1}{2} 2 sin (u)

u = 2 x

代回

= 2 \frac{1}{2} 2 sin (2 x)

化简

2 \frac{1}{2} 2 sin (2 x) : sin (2 x)

= sin (2 x)

解答补常数

= sin (2 x) + C

作图

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