Evaluate the integrals in Exercises 31–56. Some integrals do n...

Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ (sin⁻¹ x)² / √(1 - x²) dx

Accepted Answer

Welcome back, everyone. Evaluate the integral inverse of sin of X, cubed divided by square root of 1 minus X squared X. A says 1/4 multiplied by 1 minus X2 cubed plus C. B 4 multiplied by 1 minus X2 cubed plus C. C14 inverts of sin of X's the power of 4 plus C. And the 4 inverse of sin of X to the power of 4 plus C. For this problem, we're going to use to use substitution. Let's suppose that you use equal to inverse of sine of X. This is very useful because if we find the derivative of you with respect to X, we can show that it is equal to 1 divided by square root of 1 minus X2. Which is basically the expression in our denominator. So now if we multiply both sides by D X, we can show that DU is equal to D X divided by square root of 1 minus X squared. And indeed in our integrant we have D X divided by that radical term. So we can now rewrite our integral as follows. In the numerator, we have you cubed. And D X divided by the radical term is simply DU. Now we have an elementary integral, we can integrate using the power rule. The integral of U cubed is to use the power of 4 divided by 4. We're going to add the constant of integration C. And I'm replacing you with the actual expression, we're going to get 1/4. Multiplied by inverse of sine of X, which is raised to the power of 4 plus a constant of integration C which corresponds to the answer choice C. Thank you for watching.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ (sin⁻¹ x)² / √(1 - x²) dx

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Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (sin⁻¹ x)² / √(1 - x²) dx