What is the integral of xsqrt(4+9x^2) ?

The integral of xsqrt(4+9x^2) is 1/27 (4+9x^2)^{3/2}+C

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integral of xsqrt(4+9x^2)

\int x 4 + 9 x^{2} d x

\frac{1}{27} (4 + 9 x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 4 + 9 x^{2} d x

Apply u-substitution

: \int \frac{u}{18} d u

= \int \frac{u}{18} d u

Take the constant out:

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

= \frac{1}{18} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{18} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 4 + 9 x^{2}

= \frac{1}{18} \cdot \frac{2}{3} (4 + 9 x^{2})^{\frac{3}{2}}

Simplify

\frac{1}{18} \cdot \frac{2}{3} (4 + 9 x^{2})^{\frac{3}{2}} : \frac{1}{27} (4 + 9 x^{2})^{\frac{3}{2}}

= \frac{1}{27} (4 + 9 x^{2})^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{27} (4 + 9 x^{2})^{\frac{3}{2}} + C

f (x) = \frac{x}{x ^{2} + 25}

\frac{\partial}{\partial y} (3 x y + e^{- x^{2} y})

f (x) = x^{2} - 3 x + 1, at x = 3

n = 2 \sum \infty \frac{1}{( 3 n - 2 ) ^{5}}

f (x) = - x^{3} + 2 x^{2} - 4, at x = - 1

What is the integral of xsqrt(4+9x^2) ?
The integral of xsqrt(4+9x^2) is 1/27 (4+9x^2)^{3/2}+C