Evaluating an Indefinite Integral of cos^4(x)sin(x)dx

In summary, an indefinite integral is a calculus operation that involves finding the antiderivative of a function. To evaluate it, you can use various techniques such as substitution, integration by parts, or trigonometric substitution. The indefinite integral of cos^4(x)sin(x)dx is (1/5)cos^5(x) + C, which can be simplified using double angle and power reducing formulas. When evaluating this indefinite integral, it is important to consider special cases such as arbitrary constants and trigonometric functions with half-angle arguments.
  • #1
sapiental
118
0
evaluate the indefinite integral cos^4(x)sin(x)dx

I tried using the half angle formula but this gives me a much more difficult integral, so i resorted to just regular substitution but am not sure if I can do this.

u = cos(x)
du = -sin(x)dx

indefinite integral -u^4du

then -1/5(u)^5 + C or -1/5(cos^5) + C

thanks!
 
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  • #2
That's correct.
 

FAQ: Evaluating an Indefinite Integral of cos^4(x)sin(x)dx

1. What is an indefinite integral?

An indefinite integral is an operation in calculus that involves finding the antiderivative of a function. It represents the set of all possible functions whose derivative is the given function.

2. How do you evaluate an indefinite integral?

To evaluate an indefinite integral, you can use various techniques such as substitution, integration by parts, or trigonometric substitution. These techniques involve manipulating the given function to make it easier to integrate and then applying integration rules.

3. What is the indefinite integral of cos^4(x)sin(x)dx?

The indefinite integral of cos^4(x)sin(x)dx is (1/5)cos^5(x) + C, where C is the constant of integration.

4. Can you simplify the indefinite integral of cos^4(x)sin(x)dx?

Yes, the indefinite integral of cos^4(x)sin(x)dx can be simplified using the double angle formula for cos^2(x) and the power reducing formula for cos^4(x).

5. Are there any special cases to consider when evaluating this indefinite integral?

Yes, when evaluating the indefinite integral of cos^4(x)sin(x)dx, it is important to note that if the limits of integration are not specified, the answer will have an arbitrary constant added to it. Additionally, if the limits of integration are specified, the answer may involve trigonometric functions with half-angle arguments.

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