What is the integral of 1/(10^5x-x^2) ?

The integral of 1/(10^5x-x^2) is 1/100000 ln|x|-1/100000 ln|x-100000|+C

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integral of 1/(10^5x-x^2)

\int \frac{1}{1 0 ^{5} x - x ^{2}} d x

\frac{1}{100000} ln ∣ x ∣ - \frac{1}{100000} ln ∣ x - 100000 ∣ + C

Solution steps

\int \frac{1}{1 0 ^{5} x - x ^{2}} d x

Simplify

= \int \frac{1}{100000 x - x ^{2}} d x

Complete the square

100000 x - x^{2} : - (x - 50000)^{2} + 5000 0^{2}

= \int \frac{1}{- ( x - 50000 ) ^{2} + 5000 0 ^{2}} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 5000 0 ^{2}} d u

= \int \frac{1}{- u ^{2} + 5000 0 ^{2}} d u

Take the partial fraction of

\frac{1}{- u ^{2} + 5000 0 ^{2}} : \frac{1}{100000 ( u + 50000 )} - \frac{1}{100000 ( u - 50000 )}

= \int \frac{1}{100000 ( u + 50000 )} - \frac{1}{100000 ( u - 50000 )} d u

Apply the Sum Rule:

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

= \int \frac{1}{100000 ( u + 50000 )} d u - \int \frac{1}{100000 ( u - 50000 )} d u

\int \frac{1}{100000 ( u + 50000 )} d u = \frac{1}{100000} ln ∣ u + 50000 ∣

\int \frac{1}{100000 ( u - 50000 )} d u = \frac{1}{100000} ln ∣ u - 50000 ∣

= \frac{1}{100000} ln ∣ u + 50000 ∣ - \frac{1}{100000} ln ∣ u - 50000 ∣

Substitute back

u = x - 50000

= \frac{1}{100000} ln ∣ x - 50000 + 50000 ∣ - \frac{1}{100000} ln ∣ x - 50000 - 50000 ∣

Simplify

\frac{1}{100000} ln ∣ x - 50000 + 50000 ∣ - \frac{1}{100000} ln ∣ x - 50000 - 50000 ∣ : \frac{1}{100000} ln ∣ x ∣ - \frac{1}{100000} ln ∣ x - 100000 ∣

= \frac{1}{100000} ln ∣ x ∣ - \frac{1}{100000} ln ∣ x - 100000 ∣

Add a constant to the solution

= \frac{1}{100000} ln ∣ x ∣ - \frac{1}{100000} ln ∣ x - 100000 ∣ + C

\int 4 x^{\frac{1}{3}} + 2 x^{- \frac{1}{3}} d x

y^{^{'}} = y (x y^{6} + 3)

2 x^{^{''}} + 40 x = 0

\frac{x - 5}{x + 5}

f (t) = (1 + e^{2 t})^{2}

What is the integral of 1/(10^5x-x^2) ?
The integral of 1/(10^5x-x^2) is 1/100000 ln|x|-1/100000 ln|x-100000|+C