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

The integral from 0 to 2 of (2x+1)(4x^2+1) is 146/3

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

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

\frac{146}{3}

Decimal

48.66666 \dots

Solution steps

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

Multiply by the conjugate of

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

= \int_{0}^{2} \frac{4 x ^{2} - 1}{2 x - 1} (4 x^{2} + 1) 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}} \frac{4 x ^{2} - 1}{2 x - 1} (4 x^{2} + 1) d x + \int_{\frac{1}{2}}^{2} \frac{4 x ^{2} - 1}{2 x - 1} (4 x^{2} + 1) d x

\int_{0}^{\frac{1}{2}} \frac{4 x ^{2} - 1}{2 x - 1} (4 x^{2} + 1) d x = \frac{25}{24}

\int_{\frac{1}{2}}^{2} \frac{4 x ^{2} - 1}{2 x - 1} (4 x^{2} + 1) d x = \frac{381}{8}

= \frac{25}{24} + \frac{381}{8}

Simplify

\frac{25}{24} + \frac{381}{8} : \frac{146}{3}

= \frac{146}{3}

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What is the integral from 0 to 2 of (2x+1)(4x^2+1) ?
The integral from 0 to 2 of (2x+1)(4x^2+1) is 146/3