Final answer to the problem
$\frac{32}{15}\sqrt{4-x^2}+\frac{4x^{2}\sqrt{4-x^2}}{15}+\frac{\frac{1}{4}x^{4}\sqrt{4-x^2}}{5}+\frac{-x^{4}\sqrt{4-x^2}}{4}-\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{-x\sqrt{4-x^2}}{8}+C_0$
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Step-by-step Solution
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- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
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Simplifying
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$\int-\sqrt{4-x^2}\left(x^3\cos\left(\frac{x}{2}\right)+\frac{1}{2}\right)dx$
Learn how to solve problems step by step online. Integrate int(2/-2(x^3cos(x/2)+1/2)(4-x^2)^(1/2))dx. Simplifying. The integral of a function times a constant (-1) is equal to the constant times the integral of the function. Rewrite the integrand \sqrt{4-x^2}\left(x^3\cos\left(\frac{x}{2}\right)+\frac{1}{2}\right) in expanded form. Expand the integral \int\left(\sqrt{4-x^2}x^3\cos\left(\frac{x}{2}\right)+\frac{1}{2}\sqrt{4-x^2}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately.
Final answer to the problem
$\frac{32}{15}\sqrt{4-x^2}+\frac{4x^{2}\sqrt{4-x^2}}{15}+\frac{\frac{1}{4}x^{4}\sqrt{4-x^2}}{5}+\frac{-x^{4}\sqrt{4-x^2}}{4}-\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{-x\sqrt{4-x^2}}{8}+C_0$
