What is the integral of (t+2)sqrt(2+3t) ?

The integral of (t+2)sqrt(2+3t) is 1/9 (2/5 (2+3t)^{5/2}+8/3 (2+3t)^{3/2})+C

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integral of (t+2)sqrt(2+3t)

\int (t + 2) 2 + 3 t d t

\frac{1}{9} (\frac{2}{5} (2 + 3 t)^{\frac{5}{2}} + \frac{8}{3} (2 + 3 t)^{\frac{3}{2}}) + C

Solution steps

\int (t + 2) 2 + 3 t d t

Apply u-substitution

: \int \frac{u ( u + 4 )}{9} d u

= \int \frac{u ( u + 4 )}{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 (u + 4) d u

Expand

u (u + 4) : u^{\frac{3}{2}} + 4 u

= \frac{1}{9} \cdot \int u^{\frac{3}{2}} + 4 u d u

Apply the Sum Rule:

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

= \frac{1}{9} (\int u^{\frac{3}{2}} d u + \int 4 u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 4 u d u = \frac{8}{3} u^{\frac{3}{2}}

= \frac{1}{9} (\frac{2}{5} u^{\frac{5}{2}} + \frac{8}{3} u^{\frac{3}{2}})

Substitute back

u = 2 + 3 t

= \frac{1}{9} (\frac{2}{5} (2 + 3 t)^{\frac{5}{2}} + \frac{8}{3} (2 + 3 t)^{\frac{3}{2}})

Add a constant to the solution

= \frac{1}{9} (\frac{2}{5} (2 + 3 t)^{\frac{5}{2}} + \frac{8}{3} (2 + 3 t)^{\frac{3}{2}}) + C

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What is the integral of (t+2)sqrt(2+3t) ?
The integral of (t+2)sqrt(2+3t) is 1/9 (2/5 (2+3t)^{5/2}+8/3 (2+3t)^{3/2})+C