Integrating cos^-2(x) after inverse substitution

C In summary, to evaluate the integral of 1/[(16-x^2)^(3/2)] using the substitution x=4sin(t), first find dx and substitute it into the equation. Then, cube both sides and plug in the values to get the simplified equation of (1/16)S [cos^-2(t)] dt. To integrate cos^-2(t), use the formula cos^-2(x) = sec^2(x) and then use the formula for the integral of sec^2(x) to get the final answer of tan(t) + C.
  • #1
mathtrickedme
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Homework Statement



Use the substitution x=4sin(t) to evaluate the integral: S 1/[(16-x^2)^(3/2)] dx

Homework Equations



x = 4sin(t)

The Attempt at a Solution



x = 4sin(t)
dx = 4cos(t) dt

4cos(t) = (16-x^2)^(1/2), i cube both sides to get

(4cos(t))^3 = (16-x^2)^(3/2), then plug in dx and denominator into the equation to get

S 4cos(t)/[(4cos(t))^3] dt simplified and constant taken out i now get
(1/16) S [cos^-2(t)] dt

how do i integrate cos^-2(t) to get a simple answer. I don't think i can use the S cos^n(t) formula because then i have to integrate sin^-4(t) afterwards.

any help would be awesome, thanks
 
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  • #2
cos^-2(x) = sec^2(x)

Integral of sec^2(x) = tan(x)
 

Related to Integrating cos^-2(x) after inverse substitution

1. What is the purpose of integrating cos^-2(x) after inverse substitution?

The purpose of integrating cos^-2(x) after inverse substitution is to simplify the integral and make it easier to solve. Inverse substitution involves replacing a variable with a function, and then integrating the resulting expression. In the case of cos^-2(x), inverse substitution helps to transform the integral into a form that can be more easily integrated.

2. How do you perform inverse substitution for cos^-2(x)?

To perform inverse substitution for cos^-2(x), you first need to identify a function that, when differentiated, yields cos^-2(x). In this case, the function would be tan(x). Then, you replace cos^-2(x) with tan(x) in the original integral and solve the resulting expression. This will help to simplify the integral and make it easier to solve.

3. Can you provide an example of integrating cos^-2(x) after inverse substitution?

Sure, an example of integrating cos^-2(x) after inverse substitution would be: ∫cos^-2(x) dx. First, we identify a function that, when differentiated, yields cos^-2(x), which in this case is tan(x). We then replace cos^-2(x) with tan(x) in the integral, giving us: ∫tan(x) dx. We can then use the substitution rule to solve the integral and simplify it further.

4. What are the benefits of integrating cos^-2(x) after inverse substitution?

The benefits of integrating cos^-2(x) after inverse substitution include simplifying the integral and making it easier to solve, as well as providing a more general solution that can be applied to a wider range of problems. Inverse substitution also helps to transform the integral into a form that can be more easily evaluated, saving time and effort in the integration process.

5. Are there any common mistakes to avoid when integrating cos^-2(x) after inverse substitution?

Yes, there are a few common mistakes to avoid when integrating cos^-2(x) after inverse substitution. These include using the wrong inverse substitution, not properly evaluating the integral, and forgetting to add the constant of integration. It is important to carefully follow the steps of inverse substitution and double check the solution to avoid these errors.

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