What is the integral of (2x+3)/(sqrt(2x+4)) ?

The integral of (2x+3)/(sqrt(2x+4)) is 1/3 (2x+4)^{3/2}-sqrt(2x+4)+C

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integral of (2x+3)/(sqrt(2x+4))

\int \frac{2 x + 3}{2 x + 4} d x

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

Solution steps

\int \frac{2 x + 3}{2 x + 4} d x

Apply u-substitution

: \int u^{2} - 1 d u

= \int u^{2} - 1 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

= \int u^{2} d u - \int 1 d u

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

\int 1 d u = u

= \frac{u ^{3}}{3} - u

Substitute back

u = 2 x + 4

= \frac{( 2 x + 4 ) ^{3}}{3} - 2 x + 4

Simplify

\frac{( 2 x + 4 ) ^{3}}{3} - 2 x + 4 : \frac{1}{3} (2 x + 4)^{\frac{3}{2}} - 2 x + 4

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

Add a constant to the solution

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

p (x) = x^{3} - 5 x^{2} - 4 x + 20

8 + cos (t^{2})

x \to \frac{3 π}{2} lim (- 2 tan (x))

\int sin^{2} (7 x) d x

\frac{\partial}{\partial x} (3 x^{3 y})

What is the integral of (2x+3)/(sqrt(2x+4)) ?
The integral of (2x+3)/(sqrt(2x+4)) is 1/3 (2x+4)^{3/2}-sqrt(2x+4)+C