Problem 14 - Calculus and its applications

Question

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int \ln (x+4) d x $$

Step-by-Step Solution

Answer

\( \int \ln(x+4) \, dx = x \ln(x+4) - x + 4 \ln|x+4| + C \).

Step 1: Choose Substitution Variables

Choose to use integration by parts to solve the integral \( \int \ln(x+4) \, dx \). Using the formula for integration by parts, let \( u = \ln(x+4) \) and \( dv = dx \).

Step 2: Compute Derivatives and Integrals

Differentiate \( u \) and integrate \( dv \). Set \( du = \frac{1}{x+4} \, dx \) and \( v = x \).

Step 3: Apply Integration by Parts Formula

Substitute into the integration by parts formula \( \int u \, dv = uv - \int v \, du \). This gives us \( \int \ln(x+4) \, dx = x \ln(x+4) - \int x \frac{1}{x+4} \, dx \).

Step 4: Simplify and Solve the Remaining Integral

Simplify \( \int x \frac{1}{x+4} \, dx \) to \( \int \left(1 - \frac{4}{x+4}\right) \, dx \) by dividing \( x \) by \( x+4 \). Thus, solve it as \( \int \, dx - \int \frac{4}{x+4} \, dx \).

Step 5: Evaluate the Simplified Integral

Evaluate \( \int \, dx = x \) and \( \int \frac{4}{x+4} \, dx = 4 \ln|x+4| \). Thus, the entire answer is \( x \ln(x+4) - x + 4 \ln|x+4| + C \).

Step 6: Check by Differentiating

Differentiate the result \( x \ln(x+4) - x + 4 \ln|x+4| + C \). Use the product rule and chain rule to confirm that the derivative simplifies back to \( \ln(x+4) \), which verifies the solution is correct.

Key Concepts

Substitution Method

The substitution method is a helpful technique in calculus that simplifies integration problems. When you have an integral that seems tough to solve directly, substitution comes to the rescue by introducing a new variable. This can make complex expressions more manageable.

• Start by identifying a part of the integrand that can be substituted with a simpler variable.
• It's important to choose a substitution that simplifies the integral enough to make it solvable.
• After choosing a substitution, change all instances of the old variable in the original integral to the new variable to complete the substitution.

For example, in the case of \[ \int \ln(x+4) \, dx \], a different technique known as integration by parts was more suitable since it involves the logarithmic function, leading to a straightforward solution.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It's a powerful tool for analyzing changes and motion, foundational to many scientific disciplines.

• Differentiation focuses on derivate functions, emphasizing how things change.
• Integration, on the other hand, deals with summing over intervals and is crucial for finding areas under curves.
• These processes are used to solve real-world problems involving motion, growth, and forces.

In the problem \[ \int \ln(x+4) \, dx \], integration by parts, a calculus technique, is used. This method is part of differential and integral calculus, illustrating the power and versatility of calculus in solving such integrals.

Definite Integral

Definite integrals represent the accumulated value over an interval, essentially giving the area under a curve between two points. They are more than just an extension of indefinite integrals, offering precise calculations with limits.

• To solve a definite integral, determine the antiderivative first.
• Next, evaluate it at the upper and lower bounds of the given interval.
• Subtract the lower bound value from the upper bound value to get the final result.

While the original exercise \[ \int \ln(x+4) \, dx \] focused on finding an indefinite integral, understanding definite integrals is crucial for applications like calculating total distance or area, providing more concrete analysis in a bounded scenario.