Problem 58 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \frac{d x}{x^{2}+2 x+2}=\) (A) \(\ln \left(x^{2}+2 x+2\right)+C\) (B) \(\ln |x+1|+C\) (C) \(\arctan (x+1)+C\) (D) \(\frac{1}{\frac{1}{3} x^{3}+x^{2}+2 x}+C\)

Step-by-Step Solution

Answer

The answer is (C) \( \arctan(x+1) + C \).

Step 1: Recognize the Integral Form

Notice that the integral is in the form of \( \int \frac{d x}{x^{2}+bx+c} \), which is often solved by completing the square in the denominator.

Step 2: Complete the Square

The quadratic expression \( x^2 + 2x + 2 \) can be rewritten by completing the square. Rewrite it as \( (x+1)^2 + 1 \) by adding and subtracting 1.

Step 3: Identify the Integral Type

The integral \( \int \frac{dx}{(x+1)^2 + 1} \) is now in the standard form \( \int \frac{dx}{a^2 + u^2} \), which corresponds to \( \frac{1}{a}\arctan\left( \frac{u}{a} \right) + C \), where \( a=1 \) and \( u=x+1 \).

Step 4: Integrate Using the Arctangent Formula

Using the formula \( \frac{1}{a} \arctan \left( \frac{u}{a} \right) + C \) for \( a=1 \) gives \( \arctan(x+1) + C \).

Step 5: Match Solution to Answer Choices

Comparing the result \( \arctan(x+1) + C \) with the options provided, we find that it matches choice (C).

Key Concepts

Completing the Square

Completing the square is a method used in mathematics to transform a quadratic expression into a perfect square trinomial plus or minus a constant. This technique is particularly useful when dealing with quadratic expressions like \(x^2 + bx + c\). By rewriting the expression in the form of a square, we can simplify the integration process. For example, take the quadratic \(x^2 + 2x + 2\).

• First, focus on the linear and quadratic terms: \(x^2 + 2x\).
• Find the number that completes the square: \(\left(\frac{2}{2}\right)^2 = 1\).
• Add and subtract this number inside the integral: \(x^2 + 2x + 1 - 1 + 2\).
• Rearrange to get a perfect square: \( (x+1)^2 + 1\).

This process converts the original expression into a more manageable form for integration, highlighting the structure needed to apply arctangent integration.

Arctangent Integration

Arctangent integration is an integral formula for functions in the form \(\int \frac{dx}{a^2 + u^2}\). This important technique allows us to tackle integrals that involve expressions resembling a squared variable plus a constant. It results in an arctangent function, written as:

• \[\frac{1}{a}\arctan\left(\frac{u}{a}\right) + C\]

In our exercise, after completing the square, the expression becomes \((x+1)^2 + 1\), fitting perfectly into this formula with \(a = 1\) and \(u = x + 1\). Applying this structured approach simplifies finding the integral:

• Substitute into the formula: \[\arctan(x+1) + C\]

Remember, this formula is particularly powerful in calculus integration when working with irreducible quadratic expressions. It's a go-to solution whenever your integral can be reorganized to match this form.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, fitting the general form \(ax^2 + bx + c\). Understanding these expressions is critical to various calculus operations. They often appear in integrations and require certain techniques for simplification, such as completing the square.

• Key components include the quadratic term \(ax^2\), the linear term \(bx\), and the constant \(c\).
• In calculus, transforming these into squares of binomials opens the door to further manipulation like integration.

Sabotaging the equation helps in identifying standard integrals such as those requiring arctangent formulas. For instance, the exercise's quadratic \(x^2 + 2x + 2\) is reshaped into a standard form through these processes. Thus, understanding these expressions' properties is essential for seamless transition in solving calculus problems.