What is the integral from s to g of 1/(1-2*m/r) ?

The integral from s to g of 1/(1-2*m/r) is

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integral from s to g of 1/(1-2*m/r)

\int_{s}^{g} \frac{1}{1 - 2 \cdot \frac{m}{r}} d r

- 2 m ln (\frac{2 ∣ m ∣}{∣ g ∣}) + 2 m ln (\frac{2 ∣ m ∣}{∣ s ∣}) + g - s + 2 m ln \frac{2 m}{g} - 1 - 2 m ln \frac{2 m}{s} - 1

Solution steps

\int_{s}^{g} \frac{1}{1 - 2 \cdot \frac{m}{r}} d r

Factor

\frac{1}{1 - 2 \frac{m}{r}}

= \int_{s}^{g} \frac{1}{1 - \frac{2 m}{r}} d r

Apply u-substitution

: \int_{\frac{2 m}{s}}^{\frac{2 m}{g}} - \frac{2 m}{u ^{2} ( - u + 1 )} d u

= \int_{\frac{2 m}{s}}^{\frac{2 m}{g}} - \frac{2 m}{u ^{2} ( - u + 1 )} d u

Take the constant out:

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

= - 2 m \cdot \int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u ^{2} ( - u + 1 )} d u

Take the partial fraction of

\frac{1}{u ^{2} ( - u + 1 )} : \frac{1}{u} + \frac{1}{u ^{2}} - \frac{1}{u - 1}

= - 2 m \cdot \int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u} + \frac{1}{u ^{2}} - \frac{1}{u - 1} 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

= - 2 m (\int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u} d u + \int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u ^{2}} d u - \int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u - 1} d u)

\int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u} d u = ln (\frac{2 ∣ m ∣}{∣ g ∣}) - ln (\frac{2 ∣ m ∣}{∣ s ∣})

\int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u ^{2}} d u = \frac{- g + s}{2 m}

\int_{\frac{2 m}{s}}^{\frac{2 m}{g}} \frac{1}{u - 1} d u = ln \frac{2 m}{g} - 1 - ln \frac{2 m}{s} - 1

= - 2 m (ln (\frac{2 ∣ m ∣}{∣ g ∣}) - ln (\frac{2 ∣ m ∣}{∣ s ∣}) + \frac{- g + s}{2 m} - (ln \frac{2 m}{g} - 1 - ln \frac{2 m}{s} - 1))

Simplify

- 2 m (ln (\frac{2 ∣ m ∣}{∣ g ∣}) - ln (\frac{2 ∣ m ∣}{∣ s ∣}) + \frac{- g + s}{2 m} - (ln \frac{2 m}{g} - 1 - ln \frac{2 m}{s} - 1)) : - 2 m ln (\frac{2 ∣ m ∣}{∣ g ∣}) + 2 m ln (\frac{2 ∣ m ∣}{∣ s ∣}) + g - s + 2 m ln \frac{2 m}{g} - 1 - 2 m ln \frac{2 m}{s} - 1

= - 2 m ln (\frac{2 ∣ m ∣}{∣ g ∣}) + 2 m ln (\frac{2 ∣ m ∣}{∣ s ∣}) + g - s + 2 m ln \frac{2 m}{g} - 1 - 2 m ln \frac{2 m}{s} - 1

\frac{d}{d x} (x^{2} - x - 10)

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

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

\frac{e ^{- s}}{s ^{2} + 4}

\frac{d}{d x} (6 sin^{3} (x))

What is the integral from s to g of 1/(1-2*m/r) ?
The integral from s to g of 1/(1-2*m/r) is