What is the integral of (1-0.4x)/((1-x)(5-x)) ?

The integral of (1-0.4x)/((1-x)(5-x)) is -0.15ln|x-1|-0.25ln|x-5|+C

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integral of (1-0.4x)/((1-x)(5-x))

\int \frac{1 - 0.4 x}{( 1 - x ) ( 5 - x )} d x

- 0.15 ln ∣ x - 1 ∣ - 0.25 ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{1 - 0.4 x}{( 1 - x ) ( 5 - x )} d x

Take the partial fraction of

\frac{1 - 0.4 x}{( 1 - x ) ( 5 - x )} : - \frac{0.15}{x - 1} - \frac{1}{4 ( x - 5 )}

= \int - \frac{0.15}{x - 1} - \frac{1}{4 ( x - 5 )} d x

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{0.15}{x - 1} d x - \int \frac{1}{4 ( x - 5 )} d x

\int \frac{0.15}{x - 1} d x = 0.15 ln ∣ x - 1 ∣

\int \frac{1}{4 ( x - 5 )} d x = \frac{1}{4} ln ∣ x - 5 ∣

= - 0.15 ln ∣ x - 1 ∣ - \frac{1}{4} ln ∣ x - 5 ∣

Simplify

= - 0.15 ln ∣ x - 1 ∣ - 0.25 ln ∣ x - 5 ∣

Add a constant to the solution

= - 0.15 ln ∣ x - 1 ∣ - 0.25 ln ∣ x - 5 ∣ + C

\int_{- \infty}^{\infty} t d t

\frac{d}{d x} ((5 - (4 - 3 x)^{2})^{3})

\frac{d y}{d x} = a^{x + y}

\int x^{4} tan^{3} (x^{5}) d x

\int sinh (5 x) d x

What is the integral of (1-0.4x)/((1-x)(5-x)) ?
The integral of (1-0.4x)/((1-x)(5-x)) is -0.15ln|x-1|-0.25ln|x-5|+C