zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of sqrt(1+cos(4θ))

解答

\int 1 + cos (4 θ) d θ

解答

\frac{1}{2} sin (2 θ) + C

求解步骤

\int 1 + cos (4 θ) d θ

使用换元积分法

= \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

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

使用三角恒等式改写

= \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

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

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

假定

cos (u) \geq 0

= \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)

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

u = 2 θ

代回

= \frac{1}{2} 2 sin (2 θ)

化简

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

= \frac{1}{2} sin (2 θ)

解答补常数

= \frac{1}{2} sin (2 θ) + C

作图

流行的例子

integral of (x^2sin^2(x))

\int (x^{2} sin^{2} (x)) d x

integral of (3x^{1/3})/(1+x^{2/3)}

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

integral of (4x^2+2+5/(x^2+1))

\int (4 x^{2} + 2 + \frac{5}{x ^{2} + 1}) d x

integral of (e^{2x}+2e^{3x}+2)e^x

\int (e^{2 x} + 2 e^{3 x} + 2) e^{x} d x

integral of x^4sqrt(8x^5-7)

\int x^{4} 8 x^{5} - 7 d x