What is the integral of 5xsqrt(x^2+2) ?

The integral of 5xsqrt(x^2+2) is 5/3 (x^2+2)^{3/2}+C

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integral of 5xsqrt(x^2+2)

\int 5 x x^{2} + 2 d x

\frac{5}{3} (x^{2} + 2)^{\frac{3}{2}} + C

Solution steps

\int 5 x x^{2} + 2 d x

Take the constant out:

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

= 5 \cdot \int x x^{2} + 2 d x

Apply u-substitution

: \int u^{2} d u

= 5 \cdot \int u^{2} d u

Apply the Power Rule

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

= 5 \cdot \frac{u ^{3}}{3}

Substitute back

u = x^{2} + 2

= 5 \cdot \frac{( x ^{2} + 2 ) ^{3}}{3}

Simplify

5 \cdot \frac{( x ^{2} + 2 ) ^{3}}{3} : \frac{5}{3} (x^{2} + 2)^{\frac{3}{2}}

= \frac{5}{3} (x^{2} + 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{5}{3} (x^{2} + 2)^{\frac{3}{2}} + C

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What is the integral of 5xsqrt(x^2+2) ?
The integral of 5xsqrt(x^2+2) is 5/3 (x^2+2)^{3/2}+C