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受欢迎的微积分 >

integral of 1/(4-9x^2)

解答

\int \frac{1}{4 - 9 x ^{2}} d x

解答

\frac{1}{12} (ln \frac{3}{2} x + 1 - ln \frac{3}{2} x - 1) + C

求解步骤

\int \frac{1}{4 - 9 x ^{2}} d x

使用换元积分法

= \int \frac{1}{6 ( - u ^{2} + 1 )} d u

提出常数:

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

= \frac{1}{6} \cdot \int \frac{1}{- u ^{2} + 1} d u

使用积分常见定则:

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

= \frac{1}{6} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

u = \frac{3}{2} x

代回

= \frac{1}{6} (\frac{ln \frac{3}{2} x + 1}{2} - \frac{ln \frac{3}{2} x - 1}{2})

化简

\frac{1}{6} (\frac{ln \frac{3}{2} x + 1}{2} - \frac{ln \frac{3}{2} x - 1}{2}) : \frac{1}{12} (ln \frac{3}{2} x + 1 - ln \frac{3}{2} x - 1)

= \frac{1}{12} (ln \frac{3}{2} x + 1 - ln \frac{3}{2} x - 1)

解答补常数

= \frac{1}{12} (ln \frac{3}{2} x + 1 - ln \frac{3}{2} x - 1) + C

作图

流行的例子

f(x)=sqrt(2x+3)

f (x) = 2 x + 3

derivative of 2/(sqrt(3-x))

\frac{d}{d x} (\frac{2}{3 - x})

derivative of sec(tan^2(x^4))

\frac{d}{d x} (sec (tan^{2} (x^{4})))

dy=(xy^2+3xy)dx

d y = (x y^{2} + 3 x y) d x

y^{''}+4y^'+4y=25cos(x)

y^{^{''}} + 4 y^{^{'}} + 4 y = 25 cos (x)