What is the integral from 0 to 1 of x^5sqrt(1+4x^2) ?

The integral from 0 to 1 of x^5sqrt(1+4x^2) is (-1+125sqrt(5))/(840)

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integral from 0 to 1 of x^5sqrt(1+4x^2)

\int_{0}^{1} x^{5} 1 + 4 x^{2} d x

\frac{- 1 + 125 5}{840}

Decimal

0.33155 \dots

Solution steps

\int_{0}^{1} x^{5} 1 + 4 x^{2} d x

Multiply by the conjugate of

1 + 4 x^{2} : \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}}

= \int_{0}^{1} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x

Find undefined (singularity) points:

x = \frac{1}{2}

If exist

b, a < b < c, f (b) =

undefined

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{\frac{1}{2}} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x + \int_{\frac{1}{2}}^{1} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x

\int_{0}^{\frac{1}{2}} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x = \frac{11 2 - 4}{3360}

\int_{\frac{1}{2}}^{1} x^{5} \frac{1 - 16 x ^{4}}{1 - 4 x ^{2}} d x = \frac{- 11 2 + 500 5}{3360}

= \frac{11 2 - 4}{3360} + \frac{- 11 2 + 500 5}{3360}

Simplify

\frac{11 2 - 4}{3360} + \frac{- 11 2 + 500 5}{3360} : \frac{- 1 + 125 5}{840}

= \frac{- 1 + 125 5}{840}

What is the integral from 0 to 1 of x^5sqrt(1+4x^2) ?
The integral from 0 to 1 of x^5sqrt(1+4x^2) is (-1+125sqrt(5))/(840)