What is the integral of 1/(sqrt(25+3x)) ?

The integral of 1/(sqrt(25+3x)) is 2/3 sqrt(25+3x)+C

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integral of 1/(sqrt(25+3x))

\int \frac{1}{25 + 3 x} d x

\frac{2}{3} 25 + 3 x + C

Solution steps

\int \frac{1}{25 + 3 x} d x

Apply u-substitution

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= \frac{1}{3} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 25 + 3 x

= \frac{1}{3} \cdot 2 (25 + 3 x)^{\frac{1}{2}}

Simplify

\frac{1}{3} \cdot 2 (25 + 3 x)^{\frac{1}{2}} : \frac{2}{3} 25 + 3 x

= \frac{2}{3} 25 + 3 x

Add a constant to the solution

= \frac{2}{3} 25 + 3 x + C

What is the integral of 1/(sqrt(25+3x)) ?
The integral of 1/(sqrt(25+3x)) is 2/3 sqrt(25+3x)+C