Evaluate the integral:

{eq}\displaystyle \int_1^9 \frac {8 + x^2}{\sqrt x} \ dx {/eq}.


Evaluate the integral:

{eq}\displaystyle \int_1^9 \frac {8 + x^2}{\sqrt x} \ dx {/eq}.

Definite Integration:

The execution of the definite integral can be split into two steps. The first step is to acquire the antiderivative equation to the integrand. This would be following similar rules to indefinite integration. Then, we evaluate the antiderivative over the limits of the integral and then take their difference.

Answer and Explanation:

We evaluate the given integral. We do this by first simplifying the expression, which entails writing the root as a fractional exponent, separating the fraction, and then applying the property of exponents,

{eq}\displaystyle \frac{x^a}{x^b} = x^{a-b} {/eq}

Afterwards, we apply the power rule of integration, which has a formula expressed as

{eq}\displaystyle \int x^a dx = \frac{x^{a+1}}{a+1} +C {/eq}

We proceed with the solution.

{eq}\begin{align} \displaystyle \int_1^9 \frac{8 + x^2}{\sqrt x}dx &=\int_1^9 \frac{8 + x^2}{x^{1/2}}dx\\[0.3cm] &=\int_1^9 \left( \frac{1}{x^{1/2}} + \frac{x^2}{x^{1/2}} \right)dx \\[0.3cm] &= \int_1^9 (x^{-1/2} + x^{2-1/2}) dx \\[0.3cm] &= \int_1^9 (x^{-1/2} + x^{3/2}) dx\\[0.3cm] &= \left( \frac{x^{1/2}}{ \frac{1}{2}} + \frac{x^{5/2}}{ \frac{5}{2}} \right)\bigg|_1^9 \\[0.3cm] &= 2(9^{1/2} - 1^{1/2}) + \frac{2}{5}(9^{5/2} - 1^{5/2}) \\[0.3cm] &= 2(3-1) + \frac{2}{5}(243-1) \\[0.3cm] &=\mathbf{ \frac{504}{5}} \end{align} {/eq}

Learn more about this topic:

Definite Integrals: Definition


Chapter 12 / Lesson 6

A definite integral is found as the limit between a line graphed from an equation, and the x-axis, either positive or negative. Learn how this limit is identified in practical examples of definite integrals.

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