Question
Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of $x$$$\int \frac{x^{3}}{\sqrt{4-x^{2}}} d x$$
Step 1
We let $x = 2\sin(\theta)$, so $dx = 2\cos(\theta)d\theta$. The integral then becomes $$\int \frac{(2\sin(\theta))^{3}}{\sqrt{4-(2\sin(\theta))^{2}}} \cdot 2\cos(\theta) d\theta = 8\int \sin^{3}(\theta)\cos(\theta) d\theta.$$ Show more…
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