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(5-3y^2))/y

解答

\int \frac{5 - 3 y ^{2}}{y} d y

解答

- 5 (\frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} + 1 - \frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} - 1 - \frac{1}{5} (5 - 3 y^{2})) + C

求解步骤

\int \frac{5 - 3 y ^{2}}{y} d y

使用换元积分法

= \int \frac{u ^{2}}{u ^{2} - 5} d u

因式分解

u^{2} - 5 : - (- u^{2} + 5)

= \int \frac{u ^{2}}{- ( - u ^{2} + 5 )} d u

提出常数:

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

= - \int \frac{u ^{2}}{- u ^{2} + 5} d u

使用换元积分法

= - \int \frac{5 v ^{2}}{- v ^{2} + 1} d v

提出常数:

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

= - 5 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - 5 \cdot \int \frac{1}{- v ^{2} + 1} - 1 d v

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 5 (\int \frac{1}{- v ^{2} + 1} d v - \int 1 d v)

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= - 5 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

代回

= - 5 \frac{ln \frac{5 - 3 y ^{2}}{5} + 1}{2} - \frac{ln \frac{5 - 3 y ^{2}}{5} - 1}{2} - \frac{5 - 3 y ^{2}}{5}

化简

- 5 \frac{ln \frac{5 - 3 y ^{2}}{5} + 1}{2} - \frac{ln \frac{5 - 3 y ^{2}}{5} - 1}{2} - \frac{5 - 3 y ^{2}}{5} : - 5 (\frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} + 1 - \frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} - 1 - \frac{1}{5} (5 - 3 y^{2}))

= - 5 (\frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} + 1 - \frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} - 1 - \frac{1}{5} (5 - 3 y^{2}))

解答补常数

= - 5 (\frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} + 1 - \frac{1}{2} ln \frac{5 - 3 y ^{2}}{5} - 1 - \frac{1}{5} (5 - 3 y^{2})) + C

作图

流行的例子

integral of cos^3(x)csc^3(x)

\int cos^{3} (x) csc^{3} (x) d x

integral of xsqrt(25x-8)

\int x 25 x - 8 d x

integral of (sec(2x)-1)^2

\int (sec (2 x) - 1)^{2} d x

integral of-x*ln(9+x^2)

\int - x \cdot ln (9 + x^{2}) d x

integral of 10\sqrt[3]{t^2}

\int 10 3 t^{2} d t