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# Integral of sin^2x: A Step-by-Step Guide

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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.

Recommended Reading: 5 FUN WAYS TO TURN EVERY CHILD INTO A “MATH PERSON”

### 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.

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#### Moonpreneur

Moonpreneur is an ed-tech company that imparts tech entrepreneurship to children aged 6 to 15. Its flagship offering, the Innovator Program, offers students a holistic learning experience that blends Technical Skills, Power Skills, and Entrepreneurial Skills with streams such as Robotics, Game Development, App Development, Advanced Math, Scratch Coding, and Book Writing & Publishing.
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