Evaluate The Integral:$\[ \int \sin^2(2x) \, Dx \\]

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Introduction

The integral of $\sin^2(2x)$ is a fundamental problem in calculus, and it has various applications in mathematics and physics. In this article, we will evaluate the integral using trigonometric identities and provide a step-by-step solution.

Background

To evaluate the integral, we need to recall some trigonometric identities. The double-angle formula for sine is given by:

$$\sin(2x) = 2\sin(x)\cos(x)$$

We can use this formula to rewrite $\sin^2(2x)$ in terms of $\sin(x)$ and $\cos(x)$.

Step 1: Rewrite $\sin^2(2x)$ using the double-angle formula

Using the double-angle formula, we can rewrite $\sin^2(2x)$ as:

$$\sin^2(2x) = (2\sin(x)\cos(x))^2 = 4\sin^2(x)\cos^2(x)$$

Step 2: Use the Pythagorean identity to rewrite $\sin^2(x)\cos^2(x)$

The Pythagorean identity states that:

$$\sin^2(x) + \cos^2(x) = 1$$

We can use this identity to rewrite $\sin^2(x)\cos^2(x)$ as:

$$\sin^2(x)\cos^2(x) = \frac{1}{4}(1 - \cos(4x))$$

Step 3: Evaluate the integral

Now we can evaluate the integral using the substitution $u = 4x$:

$$\int \sin^2(2x) \, dx = \int \frac{1}{4}(1 - \cos(4x)) \, dx$$

$$= \frac{1}{4} \int (1 - \cos(4x)) \, dx$$

$$= \frac{1}{4} \left[ x - \frac{1}{4} \sin(4x) \right] + C$$

Conclusion

In this article, we evaluated the integral $\int \sin^2(2x) \, dx$ using trigonometric identities and provided a step-by-step solution. The final answer is $\frac{1}{4} \left[ x - \frac{1}{4} \sin(4x) \right] + C$.

Applications

The integral $\int \sin^2(2x) \, dx$ has various applications in mathematics and physics. For example, it can be used to solve problems involving the motion of a pendulum or the vibration of a string.

Further Reading

For further reading on trigonometric integrals, we recommend the following resources:

• "Trigonometry" by I.M. Gelfand and M.L. Gelfand
• "Calculus" by Michael Spivak
• "Trigonometric Integrals" by Wolfram MathWorld

References

• "Trigonometry" by I.M. Gelfand and M.L. Gelfand
• "Calculus" by Michael Spivak
• "Trigonometric Integrals" by Wolfram MathWorld

Final Answer

The final answer is $\boxed{\frac{1}{4} \left[ x - \frac{1}{4} \sin(4x) \right] + C}$.

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Introduction

In our previous article, we evaluated the integral $\int \sin^2(2x) \, dx$ using trigonometric identities. In this article, we will answer some frequently asked questions related to this integral.

Q1: What is the final answer to the integral $\int \sin^2(2x) \, dx$?

A1: The final answer to the integral $\int \sin^2(2x) \, dx$ is $\frac{1}{4} \left[ x - \frac{1}{4} \sin(4x) \right] + C$.

Q2: How do I apply the double-angle formula to rewrite $\sin^2(2x)$?

A2: To apply the double-angle formula, you need to use the identity $\sin(2x) = 2\sin(x)\cos(x)$. Then, you can square both sides of the equation to get $\sin^2(2x) = (2\sin(x)\cos(x))^2 = 4\sin^2(x)\cos^2(x)$.

Q3: What is the Pythagorean identity, and how do I use it to rewrite $\sin^2(x)\cos^2(x)$?

A3: The Pythagorean identity states that $\sin^2(x) + \cos^2(x) = 1$. You can use this identity to rewrite $\sin^2(x)\cos^2(x)$ as $\frac{1}{4}(1 - \cos(4x))$.

Q4: How do I evaluate the integral $\int \sin^2(2x) \, dx$ using the substitution $u = 4x$?

A4: To evaluate the integral using the substitution $u = 4x$, you need to replace $x$ with $\frac{u}{4}$ and $dx$ with $\frac{du}{4}$. Then, you can integrate the resulting expression to get the final answer.

Q5: What are some applications of the integral $\int \sin^2(2x) \, dx$?

A5: The integral $\int \sin^2(2x) \, dx$ has various applications in mathematics and physics, such as solving problems involving the motion of a pendulum or the vibration of a string.

Q6: Where can I find more information on trigonometric integrals?

A6: You can find more information on trigonometric integrals in textbooks such as "Trigonometry" by I.M. Gelfand and M.L. Gelfand, "Calculus" by Michael Spivak, and online resources such as Wolfram MathWorld.

Q7: What is the significance of the constant $C$ in the final answer?

A7: The constant $C$ is the constant of integration, which represents the family of antiderivatives of the function. It is an arbitrary constant that can take any value.

Q8: How do I check my work when evaluating the integral $\int \sin^2(2x) \, dx$?

A8: To check your work, you can use the following steps:

• Verify that you have applied the double-angle formula correctly.
• Check that you have used the Pythagorean identity correctly.
• Evaluate the integral using the substitution $u = 4x$ and verify that you get the correct answer.
• Check that you have included the constant of integration $C$ in the final answer.

Conclusion

In this article, we answered some frequently asked questions related to the integral $\int \sin^2(2x) \, dx$. We hope that this article has been helpful in clarifying any doubts you may have had about this integral.

Further Reading

For further reading on trigonometric integrals, we recommend the following resources:

• "Trigonometry" by I.M. Gelfand and M.L. Gelfand
• "Calculus" by Michael Spivak
• "Trigonometric Integrals" by Wolfram MathWorld

References

• "Trigonometry" by I.M. Gelfand and M.L. Gelfand
• "Calculus" by Michael Spivak
• "Trigonometric Integrals" by Wolfram MathWorld

Final Answer

The final answer is $\boxed{\frac{1}{4} \left[ x - \frac{1}{4} \sin(4x) \right] + C}$.