Evaluate the integrals in Exercises 53–76.
75. ∫y dy/√(1...

Question

Evaluate the integrals in Exercises 53–76.

75. ∫y dy/√(1-y^4)

Accepted Answer

Welcome back everyone. Evaluate the integral YDY divided by the square root of 1 minus 9Y's the power of 4. Let's begin by applying the properties of exponents. We have integral of y Dy, and in the denominator we have square root of 1 minus 9 y to the power of 4. Now if we apply the properties of exponents, we can rewrite the final term as 3 y squared. Squared And from here we're going to apply the substitution. Let's suppose that u is equal to 3 y 2, and let's remember that the next step is to find the derivative of u with respect to y. That's the derivative of 3 y 2, which is 6 y using the power rule. Now notice that in the numerator we have y Dy. So what we can do is multiply both sides by Dy and divide both sides by 6, so that 1/6 duu is going to be equal to y Dy, and this allows us to rewrite the integral in terms of u and duu. YDY in the numerator becomes 1/6 DU. And in the denominator we get the square root of 1 minus u 2. We can factor out 1/6. And integrate d u divided by square root of 1 minus u squared. Notice that this is an elementary integral and its value is inverse of sin of u. So we get 1/6 inverse of sin of u plus a constant of integration c because that's an indefinite integral. And finally, let's replace you with 3 y 2 to get 1/6. Inverse of sine of 3 y squared. Plus C. That's our final answer and thank you for watching.

Evaluate the integrals in Exercises 53–76.75. ∫y dy/√(1-y^4)

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Evaluate the integrals in Exercises 53–76.
75. ∫y dy/√(1-y^4)