What is the integral of 5/(4-sqrt(3-z)) ?

The integral of 5/(4-sqrt(3-z)) is 10(-4+sqrt(3-z)+4ln|4-sqrt(3-z)|)+C

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integral of 5/(4-sqrt(3-z))

\int \frac{5}{4 - 3 - z} d z

10 (- 4 + 3 - z + 4 ln 4 - 3 - z) + C

Solution steps

\int \frac{5}{4 - 3 - z} d z

Take the constant out:

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

= 5 \cdot \int \frac{1}{4 - 3 - z} d z

Apply u-substitution

: \int - \frac{1}{4 - u} d u

= 5 \cdot \int - \frac{1}{4 - u} d u

Take the constant out:

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

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

Apply u-substitution

: \int \frac{2 v}{4 - v} d v

= 5 (- \int \frac{2 v}{4 - v} d v)

Take the constant out:

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

= 5 (- 2 \cdot \int \frac{v}{4 - v} d v)

Apply u-substitution

: \int - \frac{- w + 4}{w} d w

= 5 (- 2 \cdot \int - \frac{- w + 4}{w} d w)

Take the constant out:

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

= 5 (- 2 (- \int \frac{- w + 4}{w} d w))

Expand

\frac{- w + 4}{w} : - 1 + \frac{4}{w}

= 5 (- 2 (- \int - 1 + \frac{4}{w} d w))

Apply the Sum Rule:

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

= 5 (- 2 (- (- \int 1 d w + \int \frac{4}{w} d w)))

\int 1 d w = w

\int \frac{4}{w} d w = 4 ln ∣ w ∣

= 5 (- 2 (- (- w + 4 ln ∣ w ∣)))

Substitute back

= 5 (- 2 (- (- (4 - 3 - z) + 4 ln 4 - 3 - z)))

Simplify

= 10 (- 4 + 3 - z + 4 ln 4 - 3 - z)

Add a constant to the solution

= 10 (- 4 + 3 - z + 4 ln 4 - 3 - z) + C

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What is the integral of 5/(4-sqrt(3-z)) ?
The integral of 5/(4-sqrt(3-z)) is 10(-4+sqrt(3-z)+4ln|4-sqrt(3-z)|)+C