Question

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{0}^{3} \ln \left(x^{2}+1\right) d x \text { (Express the answer in exact form.) } $$

Short answer

The integral evaluates to \( 3\ln(10) - 6 + 2\tan^{-1}(3) \).

Step-by-step solution

Recognize the Integral

We need to find the definite integral of the function \( \ln(x^2 + 1) \) from 0 to 3. This problem requires understanding of integration techniques and properties of logarithmic functions.

Choose an Integration Method

The integrand \( \ln(x^2 + 1) \) doesn't have a straightforward antiderivative, making it a candidate for numerical integration, or advanced techniques like integration by parts, if applicable.

Integration by Parts Setup

Integration by parts can be applied here: \( \int u \, dv = uv - \int v \, du \). Let \( u = \ln(x^2 + 1) \) and \( dv = dx \). Consequently, \( du = \frac{2x}{x^2+1} dx \) and \( v = x \).

Apply Integration by Parts

Using the integration by parts formula, we have: \[\int \ln(x^2 + 1) \, dx = x \ln(x^2 + 1) - \int \frac{2x^2}{x^2+1} \, dx.\]

Simplify the Remaining Integral

Simplify the integral: \[ \int \frac{2x^2}{x^2+1} \, dx = \int (2 - \frac{2}{x^2+1}) \, dx = 2x - 2 \tan^{-1}(x). \]

Compute the Definite Integral

Now plug in the limits of integration (0 to 3) into the expression: \[ \left[ x \ln(x^2 + 1) - 2x + 2\tan^{-1}(x) \right]_0^3. \]

Evaluate at the Upper Limit

For \( x = 3 \), compute: \( 3\ln(10) - 6 + 2\tan^{-1}(3). \)

Evaluate at the Lower Limit

For \( x = 0 \), compute: \( 0. \)

Compute the Final Answer

Subtract the lower limit from the upper limit to get the definite integral: \( 3\ln(10) - 6 + 2\tan^{-1}(3). \)

Confirm Using a Graphing Utility

Verify the solution by evaluating the integral using a graphing utility to ensure accuracy.

Key concepts

Integration by Parts

Integration by parts is a powerful tool to find integrals that aren't easily solvable by basic techniques. If you encounter an integral that looks complicated, this method might work for you. The integration by parts formula is:\[ \int u \, dv = uv - \int v \, du \]This formula comes from the product rule for differentiation. Here's how it works:

• Choose \( u \) and \( dv \) from your integrand. Usually, \( u \) is a function that becomes simpler when differentiated, and \( dv \) is easy to integrate.
• After choosing \( u = \ln(x^2 + 1) \) and \( dv = dx \), calculate \( du \) and \( v \): \( du = \frac{2x}{x^2+1} \, dx \) and \( v = x \).
• Substitute these into the formula to compute your integral.

The method simplifies complex integrals into more manageable parts. Don't forget: practice makes perfect!

Logarithmic Functions

Logarithmic functions, like \( \ln(x) \), are inverse functions of exponentials. They're great for modeling growth and decay processes. Understanding their properties makes solving integrals involving logs much simpler.For instance, the derivative of \( \ln(x) \) is \( \frac{1}{x} \), which you observe when computing \( u \, dv \) for integration by parts. Here are some key properties:

• Log rules, such as \( \ln(ab) = \ln(a) + \ln(b) \), help simplify expressions.
• They can transform multiplication into addition, making integrations more approachable.

In our integral \( \ln(x^2 + 1) \), these properties aren't directly manipulated, but knowledge of logs still aids in setting up calculations. Remember: in integrals, sometimes the hint lies in the details of the function!

Numerical Integration

Sometimes, integrals don't yield simple antiderivatives, making numerical methods valuable for approximation. These techniques are crucial when exact solutions are hard to obtain, especially without technology. Here are some common methods:

• **Trapezoidal Rule:** It estimates the area under a curve using trapezoids, effective for smooth functions.
• **Simpson's Rule:** Utilizes parabolic arcs, offering improved accuracy over trapezoidal, particularly for functions that change curvature.

In this exercise, computing the exact form involved integration by parts. However, using a graphing utility for numerical integration provides a way to verify the solution quickly. It's usual in applied settings to rely on these methods to check complex analytical calculations.