The integrals in Exercises 1–34 converge. Evaluate the integra...

Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ (4r dr) / √(1 − r⁴)

Accepted Answer

Welcome back everyone. Evaluate the improper integral from 0 to 1 of 2 x dx divided by square root of 1 minus. X to the power of 4 for this problem, let's go ahead and rewrite the integral from 0 to 1 of 2 x divided by the square root of 1 minus x to the power of 4 followed by dx. What we can do is simply use the limit definition. And specifically we have to notice that our function is undefined when x is equal to 1 at the upper limit. So we're going to replace 1 with b, and the lower limit is going to be 0, right? We're going to show that b approaches 1, and we're going to evaluate the integral from 0 to b of 2 x divided by the square root of. 1 minus x to the power of 4 d x. And now let's focus on the integral. We're going to use u substitution. Let's suppose that u is equal to x squared. So that duu divided by dx is the derivative of x 2, which is 2 x. And if we multiply both sides by dx, we can show that duu is equal to 2 x dx, which is our enumerator. Now let's change the limits of integration. The lower one is going to be 02, which is 0, and the upper one is going to be equal to. B squared. Well then, so now, if we label our integral with i, i is going to be equal to limit. As B approaches 1. Of the integral from 0 to b squared of d u divided by The square root of 1 minus x to the power of 4 is basically u squared, right? Notice that this is an elementary integral. Its value is inverse of sin of u, so we get the limit as b approaches 1. Of inverse of sine of u evaluated from 0 up to b squared. And now we can apply the fundamental theorem of calculus part 2 to get the limit as B approaches 1. Of inverse of sin of b 2 minus inverse of sin of 0. Now inverse of sin of 0 is 0. And we can evaluate inverse of sin of 1 squared, which is inverse of sin of 1, and that's pi divided by 2, which is our final answer for this problem. Thank you for watching.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀¹ (4r dr) / √(1 − r⁴)

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The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (4r dr) / √(1 − r⁴)