"Evaluate to the integral. ∫14(x2+x)dx" solved and explained

Nefissi8u

Answered question

2021-11-07

Evaluate to the integral.

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

Tamara Donohue

Beginner2021-11-08Added 11 answers

Step 1
Given integral,

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

Step 2
Now evaluating the given integral,

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

= {(\frac{x^{2 + 1}}{2 + 1} + \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1})}_{1}^{4}

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

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

= \frac{4^{3}}{3} + \frac{2}{3} {(4)}^{\frac{3}{2}} - \frac{1^{3}}{3} - \frac{2}{3} {(1)}^{\frac{3}{2}}

= \frac{64}{3} + \frac{16}{3} - \frac{1}{3} - \frac{2}{3}

= \frac{64 + 16 - 1 - 2}{3}

= \frac{77}{3}

Step 3
Final answer:

\int_{1}^{4} (x^{2} + \sqrt{x}) d x = \frac{77}{3}

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