What is the integral of 4/(1+sqrt(5x)) ?

The integral of 4/(1+sqrt(5x)) is 4(2/(sqrt(5))sqrt(x)-2/5 ln|sqrt(5)sqrt(x)+1|+3/5)+C

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integral of 4/(1+sqrt(5x))

\int \frac{4}{1 + 5 x} d x

4 (\frac{2}{5} x - \frac{2}{5} ln 5 x + 1 + \frac{3}{5}) + C

Solution steps

\int \frac{4}{1 + 5 x} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{1 + 5 x} d x

\frac{1}{1 + 5 x} = - \frac{5 x}{1 + 5 x} + 1

= 4 \cdot \int - \frac{5 x}{1 + 5 x} + 1 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

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

\int \frac{5 x}{1 + 5 x} d x = \frac{1}{5} (5 x - 25 x + 2 ln 5 x + 1 - 3)

\int 1 d x = x

= 4 (- \frac{1}{5} (5 x - 25 x + 2 ln 5 x + 1 - 3) + x)

Expand

- \frac{1}{5} (5 x - 25 x + 2 ln 5 x + 1 - 3) + x : \frac{2}{5} x - \frac{2}{5} ln 5 x + 1 + \frac{3}{5}

= 4 (\frac{2}{5} x - \frac{2}{5} ln 5 x + 1 + \frac{3}{5})

Add a constant to the solution

= 4 (\frac{2}{5} x - \frac{2}{5} ln 5 x + 1 + \frac{3}{5}) + C

\frac{d y}{d x} = - (1 + y)^{4}, y (1) = 2

\frac{1}{2} (e^{x} + e^{- x})

\int 4 8 - 3 x d x

12 arctan (s)

\int_{1}^{2} \frac{2}{( x - 5 ) ^{3}} d x

What is the integral of 4/(1+sqrt(5x)) ?
The integral of 4/(1+sqrt(5x)) is 4(2/(sqrt(5))sqrt(x)-2/5 ln|sqrt(5)sqrt(x)+1|+3/5)+C