Find $\int \frac{3 \sin X}{1-\sin^2 X} \, Dx$.

Introduction

In this article, we will delve into the world of calculus and explore the process of solving a specific integral involving trigonometric functions. The integral in question is $\int \frac{3 \sin x}{1-\sin^2 x} \, dx$. We will break down the solution step by step, using various mathematical techniques and identities to simplify the expression and arrive at the final answer.

Understanding the Integral

The given integral involves the trigonometric function $\sin x$ in the numerator and the expression $1-\sin^2 x$ in the denominator. To begin solving this integral, we need to recognize that the denominator can be simplified using the Pythagorean identity, which states that $\sin^2 x + \cos^2 x = 1$. By rearranging this identity, we can express the denominator as $\cos^2 x$.

Applying the Pythagorean Identity

Using the Pythagorean identity, we can rewrite the integral as $\int \frac{3 \sin x}{\cos^2 x} \, dx$. This simplification allows us to introduce a new trigonometric function, the tangent, which is defined as $\tan x = \frac{\sin x}{\cos x}$. By substituting this expression into the integral, we can rewrite it as $\int \frac{3 \tan x}{\cos^2 x} \, dx$.

Simplifying the Integral

Now that we have introduced the tangent function, we can simplify the integral further by recognizing that $\cos^2 x$ can be expressed as $1 - \sin^2 x$. Substituting this expression into the integral, we get $\int \frac{3 \tan x}{1 - \sin^2 x} \, dx$. This simplification allows us to rewrite the integral in terms of the tangent function and the sine function.

Using Trigonometric Identities

To further simplify the integral, we can use the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$. By substituting this expression into the integral, we get $\int \frac{3 \frac{\sin x}{\cos x}}{1 - \sin^2 x} \, dx$. This simplification allows us to rewrite the integral in terms of the sine function and the cosine function.

Applying the Substitution Method

To solve the integral, we can use the substitution method. Let's substitute $u = \sin x$, which implies that $du = \cos x \, dx$. By making this substitution, we can rewrite the integral as $\int \frac{3 u}{1 - u^2} \, du$. This simplification allows us to solve the integral using standard integration techniques.

Solving the Integral

Now that we have simplified the integral, we can solve it using standard integration techniques. The integral $\int \frac{3 u}{1 - u^2} \, du$ can be solved using the substitution method. Let's substitute $v = 1 - u^2$, which implies that $dv = -2u \, du$. By making this substitution, we can rewrite the integral as $\int \frac{-3}{2v} \, dv$. This simplification allows us to solve the integral using standard integration techniques.

Evaluating the Integral

To evaluate the integral, we can use the fundamental theorem of calculus. The integral $\int \frac{-3}{2v} \, dv$ can be evaluated as $\frac{-3}{2} \ln |v| + C$. By substituting back $v = 1 - u^2$, we get $\frac{-3}{2} \ln |1 - u^2| + C$. Finally, by substituting back $u = \sin x$, we get $\frac{-3}{2} \ln |1 - \sin^2 x| + C$.

Conclusion

In this article, we have solved the integral $\int \frac{3 \sin x}{1-\sin^2 x} \, dx$ using various mathematical techniques and identities. We have broken down the solution step by step, using the Pythagorean identity, the tangent function, and the substitution method to simplify the expression and arrive at the final answer. The solution to the integral is $\frac{-3}{2} \ln |1 - \sin^2 x| + C$. This result demonstrates the power of mathematical techniques and identities in solving complex integrals.

Final Answer

The final answer to the integral $\int \frac{3 \sin x}{1-\sin^2 x} \, dx$ is $\boxed{\frac{-3}{2} \ln |1 - \sin^2 x| + C}$.

Introduction

In our previous article, we solved the integral $\int \frac{3 \sin x}{1-\sin^2 x} \, dx$ using various mathematical techniques and identities. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states $\sin^2 x + \cos^2 x = 1$. This identity can be used to simplify expressions involving trigonometric functions.

Q: How did you simplify the integral using the Pythagorean identity?

A: We simplified the integral by recognizing that the denominator $1-\sin^2 x$ can be expressed as $\cos^2 x$ using the Pythagorean identity. This allowed us to rewrite the integral as $\int \frac{3 \sin x}{\cos^2 x} \, dx$.

Q: What is the tangent function?

A: The tangent function is a trigonometric function defined as $\tan x = \frac{\sin x}{\cos x}$. We used this function to simplify the integral further.

Q: How did you use the substitution method to solve the integral?

A: We used the substitution method by letting $u = \sin x$, which implies that $du = \cos x \, dx$. This allowed us to rewrite the integral as $\int \frac{3 u}{1 - u^2} \, du$.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus is a fundamental concept in calculus that states that differentiation and integration are inverse processes. We used this theorem to evaluate the integral.

Q: How did you evaluate the integral using the fundamental theorem of calculus?

A: We evaluated the integral by recognizing that $\int \frac{-3}{2v} \, dv = \frac{-3}{2} \ln |v| + C$. By substituting back $v = 1 - u^2$, we got $\frac{-3}{2} \ln |1 - u^2| + C$. Finally, by substituting back $u = \sin x$, we got $\frac{-3}{2} \ln |1 - \sin^2 x| + C$.

Q: What is the final answer to the integral?

A: The final answer to the integral $\int \frac{3 \sin x}{1-\sin^2 x} \, dx$ is $\boxed{\frac{-3}{2} \ln |1 - \sin^2 x| + C}$.

Q: Can you provide more examples of using the Pythagorean identity to simplify integrals?

A: Yes, we can provide more examples. For instance, consider the integral $\int \frac{2 \sin x}{\cos^2 x} \, dx$. We can simplify this integral using the Pythagorean identity as $\int \frac{2 \sin x}{1 - \sin^2 x} \, dx$. By following the same steps as before, we can solve this integral.

Q: Can you provide more examples of using the substitution method to solve integrals?

A: Yes, we can provide more examples. For instance, consider the integral $\int \frac{4 \sin x}{\cos^2 x} \, dx$. We can simplify this integral using the substitution method by letting $u = \sin x$, which implies that $du = \cos x \, dx$. This allows us to rewrite the integral as $\int \frac{4 u}{1 - u^2} \, du$. By following the same steps as before, we can solve this integral.

Conclusion

In this article, we have provided a Q&A section to address any questions or concerns that readers may have regarding the solution to the integral $\int \frac{3 \sin x}{1-\sin^2 x} \, dx$. We have covered various topics, including the Pythagorean identity, the tangent function, the substitution method, and the fundamental theorem of calculus. We hope that this Q&A section has been helpful in clarifying any doubts or questions that readers may have.