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 1/(sec^4(1/4 x))

解答

\int \frac{1}{sec ^{4} ( \frac{1}{4} x )} d x

解答

4 (cos^{3} (\frac{x}{4}) sin (\frac{x}{4}) + \frac{3}{32} (x - sin (x))) + C

求解步骤

\int \frac{1}{sec ^{4} ( \frac{1}{4} x )} d x

化简

\frac{1}{sec ^{4} ( \frac{1}{4} x )} : cos^{4} (\frac{x}{4})

= \int cos^{4} (\frac{x}{4}) d x

使用换元积分法

= \int cos^{4} (u) \cdot 4 d u

提出常数:

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

= 4 \cdot \int cos^{4} (u) d u

化简

cos^{4} (u) : cos^{3} (u) cos (u)

= 4 \cdot \int cos^{3} (u) cos (u) d u

使用分布积分法

= 4 (cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u)

\int - 3 cos^{2} (u) sin^{2} (u) d u = - \frac{3}{8} (u - \frac{1}{4} sin (4 u))

= 4 (cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u))))

u = \frac{x}{4}

代回

= 4 (cos^{3} (\frac{x}{4}) sin (\frac{x}{4}) - (- \frac{3}{8} (\frac{x}{4} - \frac{1}{4} sin (4 \cdot \frac{x}{4}))))

化简

4 (cos^{3} (\frac{x}{4}) sin (\frac{x}{4}) - (- \frac{3}{8} (\frac{x}{4} - \frac{1}{4} sin (4 \cdot \frac{x}{4})))) : 4 (cos^{3} (\frac{x}{4}) sin (\frac{x}{4}) + \frac{3}{32} (x - sin (x)))

= 4 (cos^{3} (\frac{x}{4}) sin (\frac{x}{4}) + \frac{3}{32} (x - sin (x)))

解答补常数

= 4 (cos^{3} (\frac{x}{4}) sin (\frac{x}{4}) + \frac{3}{32} (x - sin (x))) + C

作图

流行的例子

t a y l or e^{x} + 0 + 5

limit as x approaches 0 of (1+4x)^{3/x}

x \to 0 lim ((1 + 4 x)^{\frac{3}{x}})

integral of ((tan(u))/4-4/7 u^6-2/u)

\int (\frac{tan ( u )}{4} - \frac{4}{7} u^{6} - \frac{2}{u}) d u

limit as x approaches a of 1/2

x \to a lim (\frac{1}{2})

derivative f(x)=(5x+4)^3

d er i v a t i v e f (x) = (5 x + 4)^{3}