To determine

To evaluate:

The integral ∫π0sin4⁡θdθ

Answer

    -38π

Explanation

1) Concept:

Use the property of the definite integral:

If a<b then  ∆x=b - an  thus a - bn = -∆x

Therefore,

∫bafxdx=-∫abfxdx

2) Given:

∫0πsin4⁡xdx=38π

3) Calculation:

By using the property of the definite integral:

The integral ∫π0sin4⁡θdθ can be written as

∫π0sin4 ⁡ θⅆθ=- ∫0πsin4⁡θdθ ……by reversing the limits of the integration

Now, using any symbol to denote the independent variable doesn’t change the value of the integral. So

∫π0sin4⁡θⅆθ=- ∫0πsin4⁡x dx

Therefore,

∫π0sin4⁡θⅆθ=-38π

Conclusion:

∫π0sin4⁡θⅆθ=-38π