What is the integral of sqrt(9x+1) ?

The integral of sqrt(9x+1) is 2/27 (9x+1)^{3/2}+C

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integral of sqrt(9x+1)

\int 9 x + 1 d x

\frac{2}{27} (9 x + 1)^{\frac{3}{2}} + C

Solution steps

\int 9 x + 1 d x

Apply u-substitution

: \int \frac{u}{9} d u

= \int \frac{u}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{9} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 9 x + 1

= \frac{1}{9} \cdot \frac{2}{3} (9 x + 1)^{\frac{3}{2}}

Simplify

\frac{1}{9} \cdot \frac{2}{3} (9 x + 1)^{\frac{3}{2}} : \frac{2}{27} (9 x + 1)^{\frac{3}{2}}

= \frac{2}{27} (9 x + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{27} (9 x + 1)^{\frac{3}{2}} + C

x \to 2 lim (\frac{4}{x})

\int_{- 3}^{3} (∣ x ∣ + 10) d x

\int 3 (x - 3) (x + 1) d x

- 3

x = y^{2} - 8 y, x = 3 y - y^{2}

What is the integral of sqrt(9x+1) ?
The integral of sqrt(9x+1) is 2/27 (9x+1)^{3/2}+C