Find the indefinite integral.
${\int }_{}^{}\frac{arcsi{n}^{4}x}{\sqrt{1-x^2}}dx$

• A. $arcsi{n}^{5}x+C$
• B. $\frac{1}{3}arcsi{n}^{3}x+C$
• C. $-\frac{1}{5}arcco{s}^{5}x+C$
• D. $\frac{1}{5}arcsi{n}^{5}x+C$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize the structure of the integral. The given integral is ∫(arcsin^4(x) / √(1 - x^2)) dx. Notice that the derivative of arcsin(x) is 1 / √(1 - x^2), which suggests that substitution involving arcsin(x) might simplify the problem.
2. Step 2: Perform substitution. Let u = arcsin(x). Then, du = 1 / √(1 - x^2) dx. This substitution transforms the integral into ∫u^4 du.
3. Step 3: Integrate the new expression. The integral ∫u^4 du is a standard power rule integral. Use the formula ∫u^n du = u^(n+1) / (n+1) + C, where n ≠ -1. Here, n = 4, so the integral becomes u^5 / 5 + C.
4. Step 4: Substitute back the original variable. Recall that u = arcsin(x). Replace u in the result to get (arcsin(x))^5 / 5 + C.
5. Step 5: Simplify the final expression. The final answer is (1/5) * (arcsin(x))^5 + C, which matches the correct answer provided in the problem.