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

The integral from 0 to 1 of 2xsqrt(x^2+1) is (2(2sqrt(2)-1))/3

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integral from 0 to 1 of 2xsqrt(x^2+1)

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

\frac{2 ( 2 2 - 1 )}{3}

Decimal

1.21895 \dots

Solution steps

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

Take the constant out:

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

= 2 \cdot \int_{0}^{1} x x^{2} + 1 d x

Apply u-substitution

: \int_{1}^{2} u^{2} d u

= 2 \cdot \int_{1}^{2} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{1}^{2}

= 2 [\frac{u ^{3}}{3}]_{1}^{2}

Compute the boundaries:

\frac{2 2}{3} - \frac{1}{3}

= 2 (\frac{2 2}{3} - \frac{1}{3})

Simplify

= \frac{2 ( 2 2 - 1 )}{3}

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What is the integral from 0 to 1 of 2xsqrt(x^2+1) ?
The integral from 0 to 1 of 2xsqrt(x^2+1) is (2(2sqrt(2)-1))/3