What is the integral of sqrt(2x+7) ?

The integral of sqrt(2x+7) is 1/3 (2x+7)^{3/2}+C

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integral of sqrt(2x+7)

\int 2 x + 7 d x

\frac{1}{3} (2 x + 7)^{\frac{3}{2}} + C

Solution steps

\int 2 x + 7 d x

Apply u-substitution

: \int \frac{u}{2} d u

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

Take the constant out:

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

= \frac{1}{2} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = 2 x + 7

= \frac{1}{2} \cdot \frac{2}{3} (2 x + 7)^{\frac{3}{2}}

Simplify

\frac{1}{2} \cdot \frac{2}{3} (2 x + 7)^{\frac{3}{2}} : \frac{1}{3} (2 x + 7)^{\frac{3}{2}}

= \frac{1}{3} (2 x + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (2 x + 7)^{\frac{3}{2}} + C

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

y = - (2 x - 2)^{\frac{1}{3}}, (- 3, 2)

\frac{1}{2} \frac{d y}{d x} = y + 1 cos (x)

y = e^{2 x^{2}}

\int (x + x e^{x}) d x

What is the integral of sqrt(2x+7) ?
The integral of sqrt(2x+7) is 1/3 (2x+7)^{3/2}+C