3 Integrations That My Teacher Gave That I Have No Idea How To Solve.

Introduction

As a student, we often encounter problems that seem impossible to solve. One of the most challenging topics in mathematics is integration, particularly when it comes to indefinite integrals. In this article, we will focus on three integrations that my teacher gave me, and I will provide a step-by-step solution to each of them. In this part, we will discuss the second integration that I have no idea how to solve.

The Second Integration

Problem

$$\int \frac{dx}{\sqrt{\sin^3 x \cos^3 x}}$$

Solution

To solve this integration, we need to use the following steps:

Step 1: Simplify the Expression

We can start by simplifying the expression inside the square root.

$$\sqrt{\sin^3 x \cos^3 x} = \sqrt{\sin^3 x} \sqrt{\cos^3 x}$$

Using the property of square roots, we can rewrite the expression as:

$$\sqrt{\sin^3 x} \sqrt{\cos^3 x} = \sin^{\frac{3}{2}} x \cos^{\frac{3}{2}} x$$

Step 2: Use the Substitution Method

Now, we can use the substitution method to simplify the integration.

Let $u = \sin x \cos x$. Then, $du = \cos^2 x \sin x dx$.

We can rewrite the integration as:

$$\int \frac{dx}{\sqrt{\sin^3 x \cos^3 x}} = \int \frac{dx}{\sqrt{u^3}}$$

Step 3: Simplify the Integration

Now, we can simplify the integration by using the property of square roots.

$$\int \frac{dx}{\sqrt{u^3}} = \int \frac{du}{u^{\frac{3}{2}}}$$

Using the power rule of integration, we can rewrite the integration as:

$$\int \frac{du}{u^{\frac{3}{2}}} = -\frac{2}{u^{\frac{1}{2}}} + C$$

Step 4: Substitute Back

Now, we can substitute back to the original variable.

$$-\frac{2}{u^{\frac{1}{2}}} + C = -\frac{2}{(\sin x \cos x)^{\frac{1}{2}}} + C$$

Using the property of square roots, we can rewrite the expression as:

$$-\frac{2}{(\sin x \cos x)^{\frac{1}{2}}} + C = -\frac{2}{\sqrt{\sin x \cos x}} + C$$

Final Answer

The final answer is:

$$-\frac{2}{\sqrt{\sin x \cos x}} + C$$

Conclusion

In this article, we discussed the second integration that my teacher gave me. We used the substitution method to simplify the integration and then used the power rule of integration to find the final answer. This integration is a great example of how to use the substitution method to simplify complex integrals.

The Third Integration

Problem

$$\int \frac{dx}{\sqrt{\sin^3 x \cos^3 x}}$$

Solution

To solve this integration, we need to use the following steps:

Step 1: Simplify the Expression

We can start by simplifying the expression inside the square root.

$$\sqrt{\sin^3 x \cos^3 x} = \sqrt{\sin^3 x} \sqrt{\cos^3 x}$$

Using the property of square roots, we can rewrite the expression as:

$$\sqrt{\sin^3 x} \sqrt{\cos^3 x} = \sin^{\frac{3}{2}} x \cos^{\frac{3}{2}} x$$

Step 2: Use the Substitution Method

Now, we can use the substitution method to simplify the integration.

Let $u = \sin x \cos x$. Then, $du = \cos^2 x \sin x dx$.

We can rewrite the integration as:

$$\int \frac{dx}{\sqrt{\sin^3 x \cos^3 x}} = \int \frac{dx}{\sqrt{u^3}}$$

Step 3: Simplify the Integration

Now, we can simplify the integration by using the property of square roots.

$$\int \frac{dx}{\sqrt{u^3}} = \int \frac{du}{u^{\frac{3}{2}}}$$

Using the power rule of integration, we can rewrite the integration as:

$$\int \frac{du}{u^{\frac{3}{2}}} = -\frac{2}{u^{\frac{1}{2}}} + C$$

Step 4: Substitute Back

Now, we can substitute back to the original variable.

$$-\frac{2}{u^{\frac{1}{2}}} + C = -\frac{2}{(\sin x \cos x)^{\frac{1}{2}}} + C$$

Using the property of square roots, we can rewrite the expression as:

$$-\frac{2}{(\sin x \cos x)^{\frac{1}{2}}} + C = -\frac{2}{\sqrt{\sin x \cos x}} + C$$

Final Answer

The final answer is:

$$-\frac{2}{\sqrt{\sin x \cos x}} + C$$

Conclusion

In this article, we discussed the third integration that my teacher gave me. We used the substitution method to simplify the integration and then used the power rule of integration to find the final answer. This integration is a great example of how to use the substitution method to simplify complex integrals.

The First Integration

Problem

$$\int \frac{dx}{\sqrt{\sin^3 x \cos^3 x}}$$

Solution

To solve this integration, we need to use the following steps:

Step 1: Simplify the Expression

We can start by simplifying the expression inside the square root.

$$\sqrt{\sin^3 x \cos^3 x} = \sqrt{\sin^3 x} \sqrt{\cos^3 x}$$

Using the property of square roots, we can rewrite the expression as:

$$\sqrt{\sin^3 x} \sqrt{\cos^3 x} = \sin^{\frac{3}{2}} x \cos^{\frac{3}{2}} x$$

Step 2: Use the Substitution Method

Now, we can use the substitution method to simplify the integration.

Let $u = \sin x \cos x$. Then, $du = \cos^2 x \sin x dx$.

We can rewrite the integration as:

$$\int \frac{dx}{\sqrt{\sin^3 x \cos^3 x}} = \int \frac{dx}{\sqrt{u^3}}$$

Step 3: Simplify the Integration

Now, we can simplify the integration by using the property of square roots.

$$\int \frac{dx}{\sqrt{u^3}} = \int \frac{du}{u^{\frac{3}{2}}}$$

Using the power rule of integration, we can rewrite the integration as:

$$\int \frac{du}{u^{\frac{3}{2}}} = -\frac{2}{u^{\frac{1}{2}}} + C$$

Step 4: Substitute Back

Now, we can substitute back to the original variable.

$$-\frac{2}{u^{\frac{1}{2}}} + C = -\frac{2}{(\sin x \cos x)^{\frac{1}{2}}} + C$$

Using the property of square roots, we can rewrite the expression as:

$$-\frac{2}{(\sin x \cos x)^{\frac{1}{2}}} + C = -\frac{2}{\sqrt{\sin x \cos x}} + C$$

Final Answer

The final answer is:

$$-\frac{2}{\sqrt{\sin x \cos x}} + C$$

Conclusion

In this article, we discussed the first integration that my teacher gave me. We used the substitution method to simplify the integration and then used the power rule of integration to find the final answer. This integration is a great example of how to use the substitution method to simplify complex integrals.

Conclusion

Introduction

In our previous article, we discussed three integrations that my teacher gave me. We used the substitution method to simplify each integration and then used the power rule of integration to find the final answer. In this article, we will answer some of the most frequently asked questions about these integrations.

Q: What is the substitution method?

A: The substitution method is a technique used to simplify complex integrals. It involves replacing a variable in the integral with a new variable, which makes the integral easier to solve.

Q: How do I choose the right substitution?

A: Choosing the right substitution is crucial in solving integrals. You need to choose a substitution that simplifies the integral and makes it easier to solve. In the case of the integrals we discussed earlier, we used the substitution $u = \sin x \cos x$ to simplify the integrals.

Q: What is the power rule of integration?

A: The power rule of integration is a rule used to integrate functions of the form $x^n$. It states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.

Q: How do I apply the power rule of integration?

A: To apply the power rule of integration, you need to identify the function that you want to integrate and then apply the rule. In the case of the integrals we discussed earlier, we applied the power rule of integration to the function $\frac{1}{u^{\frac{3}{2}}}$.

Q: What is the final answer to the first integration?

A: The final answer to the first integration is:

$$-\frac{2}{\sqrt{\sin x \cos x}} + C$$

Q: What is the final answer to the second integration?

A: The final answer to the second integration is:

$$-\frac{2}{\sqrt{\sin x \cos x}} + C$$

Q: What is the final answer to the third integration?

A: The final answer to the third integration is:

$$-\frac{2}{\sqrt{\sin x \cos x}} + C$$

Q: How do I know which substitution to use?

A: Choosing the right substitution is crucial in solving integrals. You need to choose a substitution that simplifies the integral and makes it easier to solve. In the case of the integrals we discussed earlier, we used the substitution $u = \sin x \cos x$ to simplify the integrals.

Q: Can I use the substitution method with any integral?

A: Yes, you can use the substitution method with any integral. However, you need to choose a substitution that simplifies the integral and makes it easier to solve.

Conclusion

In this article, we answered some of the most frequently asked questions about the integrations we discussed earlier. We also provided some tips on how to choose the right substitution and how to apply the power rule of integration. We hope that this article has been helpful in understanding the substitution method and the power rule of integration.

Additional Resources

If you want to learn more about the substitution method and the power rule of integration, we recommend the following resources:

• Mathway: A online math problem solver that can help you solve integrals.
• Wolfram Alpha: A online calculator that can help you solve integrals.
• Khan Academy: A online learning platform that provides video lessons on math and other subjects.

We hope that this article has been helpful in understanding the substitution method and the power rule of integration. If you have any further questions, please don't hesitate to ask.