integral of (2-sqrt(x))/(4-x)

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∫ 2−√x4−x dx

Solution

−4(12 ln|√x2 +1|−12 ln|√x2 −1|−12 √x)−2ln|x−4|+C

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∫ 2−√x4−x dx

2−√x4−x =(2−√x)(4+x)(4)²−(x)²

=∫ (2−√x)(4+x)4²−x² dx

Simplify (2−√x)(4+x)4²−x² : √x−2x−4

=∫ √x−2x−4 dx

Expand √x−2x−4 : √xx−4 −2x−4

Apply the Sum Rule: ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx

=∫ √xx−4 dx−∫ 2x−4 dx

∫ √xx−4 dx=−4(12 ln|√x2 +1|−12 ln|√x2 −1|−12 √x)

∫ 2x−4 dx=2ln|x−4|

=−4(12 ln|√x2 +1|−12 ln|√x2 −1|−12 √x)−2ln|x−4|

Add a constant to the solution

=−4(12 ln|√x2 +1|−12 ln|√x2 −1|−12 √x)−2ln|x−4|+C

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integral of (2-sqrt(x))/(4-x)

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