What is the integral of 3xsqrt(5+x^2) ?

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

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integral of 3xsqrt(5+x^2)

\int 3 x 5 + x^{2} d x

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

Solution steps

\int 3 x 5 + x^{2} d x

Take the constant out:

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

= 3 \cdot \int x 5 + x^{2} d x

Apply u-substitution

: \int u^{2} d u

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

Apply the Power Rule

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

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

Substitute back

u = 5 + x^{2}

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

Simplify

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

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

Add a constant to the solution

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

\int cos (2 y) d y

\int \frac{1}{x x} d x

x \to 1 lim (2 x^{2} - 4 x + 2)

\frac{\partial}{\partial y} (\frac{3 y}{x ^{5}})

y^{^{''}} - 3 y^{^{'}} + 4 y = x e^{x}

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