Integral of 5sin^(3/2)x*cosx dx

In summary, the formula for the integral of 5sin^(3/2)x*cosx dx is ∫5sin^(3/2)x*cosx dx = -2/3cos^2x + C. To solve this integral, you can use the trigonometric identity sin^2x = 1/2(1 - cos2x) and the power rule. The integral can also be simplified to ∫5sin^(3/2)x*cosx dx = -2/3cos^2x + C by using the trigonometric identity and the power rule. The constant C represents the family of solutions to the integral, and it can be applied in real-world scenarios involving areas under curves, motion,
  • #1
fiziksfun
78
0
I'm having trouble with this integral:

integral of 5sin^(3/2)x*cosx dx

any suggestions? i tried integration by substitution but it didnt work.
do i need to do integration by part?? help!
 
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  • #2
fiziksfun said:
I'm having trouble with this integral:

integral of 5sin^(3/2)x*cosx dx

any suggestions? i tried integration by substitution but it didnt work.
do i need to do integration by part?? help!

Hi fiziksfun! :smile:

Do you mean ∫5[(sinx)^(3/2)]cosx dx ?

Because, if you did the obvious substitution, it should have worked :confused:
 

1. What is the formula for the integral of 5sin^(3/2)x*cosx dx?

The formula for the integral of 5sin^(3/2)x*cosx dx is ∫5sin^(3/2)x*cosx dx = -2/3cos^2x + C.

2. How do you solve the integral of 5sin^(3/2)x*cosx dx?

To solve the integral of 5sin^(3/2)x*cosx dx, you can use the trigonometric identity sin^2x = 1/2(1 - cos2x) to rewrite the integral as ∫5sin^(3/2)x*cosx dx = -5/6∫cos2x - cosx dx. Then, you can use the power rule and substitution method to solve the integral.

3. Can the integral of 5sin^(3/2)x*cosx dx be simplified?

Yes, the integral of 5sin^(3/2)x*cosx dx can be simplified to ∫5sin^(3/2)x*cosx dx = -2/3cos^2x + C by using the trigonometric identity sin^2x = 1/2(1 - cos2x) and the power rule.

4. What is the significance of the constant C in the integral of 5sin^(3/2)x*cosx dx?

The constant C in the integral of 5sin^(3/2)x*cosx dx represents the family of solutions to the integral. This means that there are infinite possible solutions to the integral, all of which differ by a constant value.

5. How can the integral of 5sin^(3/2)x*cosx dx be applied in real-world scenarios?

The integral of 5sin^(3/2)x*cosx dx can be applied in real-world scenarios that involve the calculation of areas under curves or the determination of the position, velocity, and acceleration of an object in motion. It can also be used in engineering, physics, and other scientific fields to model and solve various problems involving trigonometric functions.

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