Evaluate the integral of x((4 - x^2))^1/2

Question:

Evaluate

$$\int x \sqrt{4 - x^2}\,dx $$

Answer and Explanation:

Rewriting the integrand in the form of power, results:

{eq}\int x \sqrt {4 - {x^2}} \,dx = \int {x{{\left( {4 - {x^2}} \right)}^{\frac{1}{2}}}} dx {/eq}

Multiplying conveniently by 2, we solve the integral:

{eq}= - \frac{1}{2}\int { - 2x{{\left( {4 - {x^2}} \right)}^{\frac{1}{2}}}} dx\\ = - \frac{1}{2}\frac{{{{\left( {4 - {x^2}} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + C\\ = - \frac{1}{3}{\left( {4 - {x^2}} \right)^{\frac{3}{2}}} + C {/eq}


Learn more about this topic:

Work as an Integral

from Physics: High School

Chapter 7 / Lesson 10
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