The integral of sin^2x is a common integral that arises in many different mathematical problems. In this article, we will show you how to integrate sin^2x using two other methods:
1. Using the double-angle formula
2. Using integration by parts
We will also discuss the importance of constant integration at the end of the integration process.
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Using the double-angle formula
The double angle formula for sin is sin(2x) = 2sinxcosx. We can use this formula to rewrite sin^2x as follows:
sin^2x = (1 – cos(2x)) / 2
Now, we can integrate using the following steps:
1. Integrate 1/2.
2. Integrate cos(2x) using the double-angle formula.
3. Add the constant of integration.
The final result is:
∫ sin^2x dx = x/2 – sin(2x)/4 + C
Using integration by parts
We can also use integration by parts to integrate sin^2x. In this case, we will let u = sinx and dv = dx. Then, du = cosx dx and v = x.
The integration by parts formula is:
∫ u dv = uv – ∫ v du
In this case, the integral becomes:
∫ sin^2x dx = sinx * x – ∫ x * cosx dx
The second integral can be integrated using the following steps:
1. Integrate x.
2. Integrate cost using the double-angle formula.
The final result is the same as the result we obtained using the double-angle formula:
∫ sin^2x dx = x/2 – sin(2x)/4 + C
The importance of constant integration
The constant of integration is essential because it represents the possible values of the integral that the given information cannot determine. In the case of sin^2x, the constant of integration can represent any potential value of sin^2x that is zero when x = 0.
Conclusion
In this article, we have shown you how to integrate sin^2x using two different methods. We have also discussed the importance of constant integration at the end of the integration process.
We hope this article has been helpful. If you have any questions, please feel free to comment below.
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