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Integral Calculus

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Question : 12 of 73

Marks: +1, -0

\int x^{2}(x^{3}+5)^{\frac{1}{2}} dx =

$\frac{2}{3}(x^{3}+5)^{\frac{3}{2}}+c$
$\frac{3}{2}(x^{3}+5)^{\frac{3}{2}}+c$
$\frac{2}{9}(x^{3}+5)^{\frac{3}{2}}+c$
none

Solution:

Concept:

Substitution method: let

u = x^3 + 5

to simplify the integral.

Explanation:

Let

u = x^3 + 5

.

Then

du = 3x^2 dx

, so

x^2 dx = \frac{du}{3}

.

The integral becomes

\int \sqrt{u} \cdot \frac{du}{3} = \frac{1}{3} \int u^{1/2} du

.

Integrate:

\frac{1}{3} \cdot \frac{u^{3/2}}{3/2} = \frac{1}{3} \cdot \frac{2}{3} u^{3/2} = \frac{2}{9} u^{3/2}

.

Substitute back:

\frac{2}{9} (x^3 + 5)^{3/2} + C

.

Answer:

\frac{2}{9}(x^{3}+5)^{\frac{3}{2}}+c

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