What is the integral of (xsqrt(4x^2+7)) ?

The integral of (xsqrt(4x^2+7)) is 1/12 (4x^2+7)^{3/2}+C

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integral of (xsqrt(4x^2+7))

\int (x 4 x^{2} + 7) d x

\frac{1}{12} (4 x^{2} + 7)^{\frac{3}{2}} + C

Solution steps

\int x 4 x^{2} + 7 d x

Apply u-substitution

: \int \frac{u}{8} d u

= \int \frac{u}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 4 x^{2} + 7

= \frac{1}{8} \cdot \frac{2}{3} (4 x^{2} + 7)^{\frac{3}{2}}

Simplify

\frac{1}{8} \cdot \frac{2}{3} (4 x^{2} + 7)^{\frac{3}{2}} : \frac{1}{12} (4 x^{2} + 7)^{\frac{3}{2}}

= \frac{1}{12} (4 x^{2} + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{12} (4 x^{2} + 7)^{\frac{3}{2}} + C

\int (sin^{2} (θ)) d θ

\int e^{x} - cos (x) d x

x^{2} + 1, 1, 3

y^{^{'}} = 2 x

y^{^{''}} - 3 y^{^{'}} = 26 sin (2 t)

What is the integral of (xsqrt(4x^2+7)) ?
The integral of (xsqrt(4x^2+7)) is 1/12 (4x^2+7)^{3/2}+C