Get the answer to "Evaluate the integrals. ∫01x(x+1)dx"

aortiH

Answered question

2021-10-06

Evaluate the integrals.

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

cyhuddwyr9

Skilled2021-10-07Added 90 answers

Consider,

\int \sqrt{x} (\sqrt{x} + 1) d x = \int (x + \sqrt{x}) d x

= \int (x + x^{\frac{1}{2}}) d x

= \frac{x^{2}}{2} + \frac{2}{3} \cdot x^{\frac{3}{2}} + C

\int_{0}^{1} \sqrt{x} (\sqrt{x} + 1) d x = {[\frac{x^{2}}{2} + \frac{2}{3} \cdot x^{\frac{3}{2}}]}_{0}^{1}

= [\frac{1^{2}}{2} + \frac{2}{3} \cdot {(1)}^{\frac{3}{2}}] - [\frac{0^{2}}{2} + \frac{2}{3} \cdot {(0)}^{\frac{3}{2}}]

= \frac{1}{2} + \frac{2}{3}

= \frac{7}{6}

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