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

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

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integral of 3sqrt(3x+1)

\int 3 3 x + 1 d x

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

Solution steps

\int 3 3 x + 1 d x

Take the constant out:

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

= 3 \cdot \int 3 x + 1 d x

Apply u-substitution

: \int \frac{u}{3} d u

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = 3 x + 1

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

Simplify

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

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

Add a constant to the solution

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

\int x (x + 2)^{4} d x

f (x) = 4 x^{3} - 12 x^{2}, at x = - 2

\int sin (9 t) d t

\int_{0}^{1} x (\frac{4}{9} x (5 - x^{2})) d x

x \to 3 lim (5 + x)

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