Question

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \frac{x\sin \left(x^2\right)}{\cos \left(x^2\right)}dx$$

Short answer

$-\frac{1}{2}\ln |\cos (x^2)|+C$

Step-by-step solution

Identify the Integral and Substitution

The given integral is $\int \frac{x\sin (x^2)}{\cos (x^2)}dx$. Notice that the integrand $\frac{\sin (x^2)}{\cos (x^2)}$ can be rewritten using the tangent identity: $\tan (x^2)$. This suggests a suitable substitution.

Determine the Substitution

To simplify the integral, let's set $u=x^2$. Then, the differential is $du=2xdx$, which gives us $dx=\frac{du}{2x}$. This substitution simplifies the trigonometric function.

Substitute and Simplify the Integral

Substitute in the integral: $\int \frac{x\tan (u)}{\cos (u)}\cdot \frac{du}{2x}=\frac{1}{2}\int \tan (u)du$. The $x$ terms cancel out, leaving a simpler integral to solve.

Integrate Using Logarithmic Identity

The integral of $\tan (u)$ is $\int \tan (u)du=-\ln |\cos (u)|+C$. Therefore, our integral becomes $-\frac{1}{2}\ln |\cos (u)|+C$.

Substitute Back to Original Variable

Replace $u$ with $x^2$ based on our substitution: $-\frac{1}{2}\ln |\cos (x^2)|+C$. Thus, the final expression of the integral in terms of compositions with logarithms is complete.

Key concepts

The Substitution Method

One of the primary techniques for solving integrals, especially those involving trigonometric functions, is the substitution method. This method involves replacing a part of the integral with a new variable to simplify the integrand. In the context of the integral $\int \frac{x\sin (x^2)}{\cos (x^2)}dx$, we identified that the term $x^2$ is effectively setting up the expression for substitution.Letting $u=x^2$ simplifies the expression significantly. The differential of $u$ in terms of $x$ is $du=2xdx$. With this substitution, you can express $dx$ as $dx=\frac{du}{2x}$, making it possible to cancel the $x$ in the numerator with the $x$ in the denominator during integration, easing the problem significantly.Using substitutions helps transform a problematic integral into a familiar, manageable form. It is particularly helpful with trigonometric integrals as it often aligns the equation into a standard formula, such as those for logarithmic integration.

Understanding Tangent Identity

The tangent identity is a crucial building block in trigonometric functions, especially with integrations involving trigonometric expressions. In our case, identifying $\tan (x^2)=\frac{\sin (x^2)}{\cos (x^2)}$ allows us to simplify the integrand of the integral.By substituting $\frac{\sin (x^2)}{\cos (x^2)}$ with $\tan (x^2)$, you often get to a form that is easier to integrate. It's like turning a complex recipe into a simple, easy-to-follow one. Remembering and using these trigonometric identities effectively can significantly streamline solving integrals and reveal paths that are not initially apparent.In this exercise, recognizing the tangent identity was the key to moving forward with the integration process efficiently.

Exploring Logarithmic Integration

Logarithmic integration is a technique used to solve integrals involving trigonometric functions expressed through identities such as tangent. In this case, we ended up with the integral $\int \tan (u)du$.A helpful formula to remember is: $\int \tan (u)du=-\ln |\cos (u)|+C$. This applies when integrating functions that break down into logarithmic forms. It's essential to get comfortable with it as it's a prevalent result in integrals involving trigonometric functions.Using logarithmic integration simplifies the calculus process and aids in solving integrals that seemed complex initially. Incorporating this technique reduces a lot of algebraic manipulations, making the integration cleaner and solving the problems more efficiently.

Variable Substitution in Integration

Variable substitution is an integral tool in calculus, often used to simplify and solve tricky integrals. By replacing the original variable with a new one, we turned our complex expression into a simpler form. In the integral $\int \frac{x\sin (x^2)}{\cos (x^2)}dx$, setting $u=x^2$ effectively transformed the trigonometric components into an easier-to-integrate form.The process involves:

• Substituting $u=x^2$ and finding $du=2xdx$.
• Expressing $dx$ as $\frac{du}{2x}$ to simplify the integral.
• Rewriting the integral in terms of $u$ to cancel out terms and solve.

This tactic reduces algebraic complexity. When facing a challenging integral, look for variable substitution as your first resort to simplify your calculations and find a cleaner, efficient pathway to the solution.