Evaluate The Following Integral Using A Trigonometric Substitution:${ \int \sqrt{2025 - 81x^2} , Dx }$Be Sure To Include Parentheses Around The Arguments Of Inverse Trigonometric Functions In Your Answer.Provide Your Answer

Mar 12, 2025 by ADMIN 225 views

Evaluate the Integral Using Trigonometric Substitution

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Introduction

In this article, we will evaluate the given integral using a trigonometric substitution. The integral is $\int \sqrt{2025 - 81x^2} \, dx$ . We will use the trigonometric substitution method to simplify the integral and find its value.

Trigonometric Substitution

The trigonometric substitution method is a technique used to evaluate integrals that involve expressions of the form $\sqrt{a^2 - x^2}$ or $\sqrt{a^2 + x^2}$ . In this case, we have the expression $\sqrt{2025 - 81x^2}$ , which can be written as $\sqrt{81(25 - x^2)}$ . This suggests that we can use the trigonometric substitution $x = 5 \tan \theta$ .

Step 1: Find the Derivative of the Substitution

To find the derivative of the substitution, we will use the chain rule. Let $u = 5 \tan \theta$ , then $du = 5 \sec^2 \theta \, d\theta$ . We can rewrite the integral in terms of $u$ and $du$ .

Step 2: Rewrite the Integral in Terms of u and du

We can rewrite the integral as $\int \sqrt{81(25 - u^2)} \, du$ . We can simplify this expression by factoring out the constant $81$ .

Step 3: Simplify the Integral

We can simplify the integral as $\int 9 \sqrt{25 - u^2} \, du$ . This is a standard integral that can be evaluated using the trigonometric substitution method.

Step 4: Evaluate the Integral

We can evaluate the integral as $\frac{9}{2} \sin^{-1} \left( \frac{u}{5} \right) + C$ . We can rewrite this expression in terms of $x$ by substituting back $u = 5 \tan \theta$ .

Step 5: Rewrite the Answer in Terms of x

We can rewrite the answer as $\frac{9}{2} \sin^{-1} \left( \frac{5 \tan \theta}{5} \right) + C$ . We can simplify this expression by canceling out the common factor of $5$ .

Step 6: Simplify the Answer

We can simplify the answer as $\frac{9}{2} \sin^{-1} (\tan \theta) + C$ . We can rewrite this expression in terms of $x$ by using the trigonometric identity $\sin^{-1} (\tan \theta) = \theta$ .

Step 7: Rewrite the Answer in Terms of x

We can rewrite the answer as $\frac{9}{2} \theta + C$ . We can rewrite this expression in terms of $x$ by using the trigonometric identity $\theta = \tan^{-1} \left( \frac{x}{5} \right)$ .

Step 8: Rewrite the Answer in Terms of x

We can rewrite the answer as $\frac{9}{2} \tan^{-1} \left( \frac{x}{5} \right) + C$ . This is the final answer to the given integral.

Conclusion

In this article, we evaluated the given integral using a trigonometric substitution. We used the trigonometric substitution method to simplify the integral and find its value. The final answer is $\frac{9}{2} \tan^{-1} \left( \frac{x}{5} \right) + C$ . This is a standard result that can be used to evaluate similar integrals.

Final Answer

The final answer to the given integral is $\boxed{\frac{9}{2} \tan^{-1} \left( \frac{x}{5} \right) + C}$ .

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Introduction

In the previous article, we evaluated the integral $\int \sqrt{2025 - 81x^2} \, dx$ using a trigonometric substitution. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the trigonometric substitution method?

A: The trigonometric substitution method is a technique used to evaluate integrals that involve expressions of the form $\sqrt{a^2 - x^2}$ or $\sqrt{a^2 + x^2}$ . This method involves substituting the variable $x$ with a trigonometric function, such as $\tan \theta$ or $\sin \theta$ , to simplify the integral.

Q: How do I choose the correct trigonometric substitution?

A: To choose the correct trigonometric substitution, you need to identify the expression inside the square root and determine which trigonometric function will simplify it. In this case, we used the substitution $x = 5 \tan \theta$ to simplify the expression $\sqrt{2025 - 81x^2}$ .

Q: What is the derivative of the trigonometric substitution?

A: The derivative of the trigonometric substitution $x = 5 \tan \theta$ is $dx = 5 \sec^2 \theta \, d\theta$ . This derivative is used to rewrite the integral in terms of $u$ and $du$ .

Q: How do I rewrite the integral in terms of u and du?

A: To rewrite the integral in terms of $u$ and $du$ , you need to substitute the variable $x$ with the trigonometric function and its derivative. In this case, we substituted $x = 5 \tan \theta$ and $dx = 5 \sec^2 \theta \, d\theta$ to rewrite the integral as $\int \sqrt{81(25 - u^2)} \, du$ .

Q: What is the final answer to the integral?

A: The final answer to the integral is $\boxed{\frac{9}{2} \tan^{-1} \left( \frac{x}{5} \right) + C}$ . This is a standard result that can be used to evaluate similar integrals.

Q: Can I use the trigonometric substitution method for other types of integrals?

A: Yes, the trigonometric substitution method can be used to evaluate other types of integrals, such as integrals involving expressions of the form $\sqrt{a^2 + x^2}$ or $\sqrt{x^2 - a^2}$ . However, the specific substitution and derivative used will depend on the expression inside the square root.

Q: What are some common mistakes to avoid when using the trigonometric substitution method?

A: Some common mistakes to avoid when using the trigonometric substitution method include:

Not identifying the correct trigonometric function to use for the substitution
Not rewriting the integral in terms of $u$ and $du$
Not using the correct derivative for the substitution
Not simplifying the integral correctly

Conclusion

In this article, we answered some frequently asked questions related to evaluating the integral using trigonometric substitution. We hope that this article has provided helpful information and guidance for those who are struggling with this topic.

Final Answer

The final answer to the integral is $\boxed{\frac{9}{2} \tan^{-1} \left( \frac{x}{5} \right) + C}$ .