To determine
To evaluate:
The integral ∫π0sin4θdθ
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:
∫0πsin4xdx=38π
3) Calculation:
By using the property of the definite integral:
The integral ∫π0sin4θdθ can be written as
∫π0sin4 θⅆθ=- ∫0πsin4θdθ ……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
∫π0sin4θⅆθ=- ∫0πsin4x dx
Therefore,
∫π0sin4θⅆθ=-38π
Conclusion:
∫π0sin4θⅆθ=-38π