Problem 15 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int x \sin \left(x^{2}\right) d x$$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{2}\cos(x^2) + C\).

Step 1: Set Up the Integral for Substitution

We need to find the indefinite integral \( \int x \sin(x^2) \, dx \). We'll use a substitution method to solve this integral.

Step 2: Determine the Appropriate Substitution

To simplify the integral, we'll let \( u = x^2 \). Then, the derivative \( du = 2x \, dx \), or equivalently, \( x \, dx = \frac{1}{2} du \).

Step 3: Substitute into the Integral

Substitute \( u \) into the integral and the differential \( x \, dx \) with \( \frac{1}{2} du \), which gives us: \[ \int x \sin(x^2) \, dx = \int \sin(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int \sin(u) \, du \].

Step 4: Solve the New Integral

The new integral is \( \frac{1}{2} \int \sin(u) \, du \). The antiderivative of \( \sin(u) \) is \( -\cos(u) \). So, we have: \[ \frac{1}{2} (-\cos(u)) + C = -\frac{1}{2} \cos(u) + C \] where \( C \) is the constant of integration.

Step 5: Substitute Back to Original Variable

Replace \( u \) with \( x^2 \) in the antiderivative: \(-\frac{1}{2} \cos(x^2) + C \). Thus, \[ \int x \sin(x^2) \, dx = -\frac{1}{2} \cos(x^2) + C \].

Step 6: Differentiate to Verify

Differentiate the result \( -\frac{1}{2} \cos(x^2) + C \) with respect to \( x \). The derivative is \(-\frac{1}{2} \cdot ( -\sin(x^2) \cdot 2x ) = x \sin(x^2) \), which matches the integrand, confirming correctness.

Key Concepts

Definite and Indefinite Integrals

Integrals are a major part of calculus and are used to find areas under curves, total accumulated values, and much more. When we talk about indefinite integrals, we refer to integrating without specific bounds. This gives us a family of functions, and the result includes a constant of integration, denoted as \( C \). Indefinite integrals have the general form:

• \( \int f(x) \, dx = F(x) + C \)

The constant \( C \) represents any constant value since differentiation causes these constants to disappear.On the other hand, definite integrals are evaluated over specific intervals \( [a, b] \). The result of a definite integral is a number that represents the net area under the curve from \( a \) to \( b \). For definite integrals, the integration process is:

• \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \)

In this exercise, we are primarily dealing with an indefinite integral, as we are focusing on finding the antiderivative of \( x \sin(x^2) \).

Substitution Method

The substitution method is a key technique to simplify integrals by changing variables. This is similar to using a change of coordinates in geometry. Through substitution, complex integrals are transformed into simpler forms that are easier to solve.To use this method effectively:

• Choose a substitution that simplifies the integral. For example, let \( u = x^2 \), making the integrand \( x \sin(x^2) \) easier to handle.
• Find the differential of the substitution. In this case, if \( u = x^2 \), then \( du = 2x \, dx \).
• Replace the original variables in the integral with new terms in \( u \): \( x \, dx = \frac{1}{2} \, du \).
• After integrating with respect to \( u \), substitute back the original variable \( x \) to find the antiderivative in terms of \( x \).

This step-by-step process transforms difficult problems into simple ones, allowing us to solve integrals that might seem perplexing at first glance.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are functions that reverse differentiation. When you find the antiderivative of a function, you are essentially looking for a new function whose derivative is the original function.For example, if you have \( f(x) = \sin(x^2) \), an antiderivative would be a function whose derivative gives us \( \sin(x^2) \). By using the substitution \( u = x^2 \), we simplify this process. In this case, the antiderivative of \( \sin(u) \) via direct integration is \( -\cos(u) + C \).Key aspects of finding antiderivatives:

• Recognize the function you are integrating. For the integral \( \int \sin(u) \, du \), the antiderivative is \( -\cos(u) + C \).
• Always include the constant of integration \( C \), as many different functions can serve as an antiderivative due to constant differences.
• After completing the substitution, revert to the original variable, as in reverting \( u \) back to \( x^2 \), so \( \int x \sin(x^2) \, dx = - \frac{1}{2} \cos(x^2) + C \).

Thus, learning how to effectively find antiderivatives aids in mastering the fundamentals of calculus.