Step-by-Step Guidance for MA 16200 Second Midterm (Calculus II Topics)

Q1. Perform polynomial long division to obtain polynomials $q(x)$ and $r(x)$ that satisfy $\frac{x^3}{x^2 + x - 3} = q(x) + \frac{r(x)}{x^2 + x - 3}$, where the degree of $r(x)$ is strictly less than 2. What is $r(x)$?

Background

Topic: Polynomial Long Division

This question tests your ability to perform polynomial long division and identify the remainder $r(x)$, ensuring its degree is less than the divisor's degree.

Key Terms and Formulas

• Dividend: $x^3$

• Divisor: $x^2 + x - 3$

• Quotient: $q(x)$

• Remainder: $r(x)$ (degree $< 2$)

• Division Algorithm: $\text{Dividend} = (\text{Divisor}) \cdot (\text{Quotient}) + \text{Remainder}$

Step-by-Step Guidance

1. Set up the long division: Divide $x^3$ by $x^2 + x - 3$.

2. Determine how many times $x^2$ fits into $x^3$ (i.e., what to multiply the divisor by to match the leading term of the dividend).

3. Multiply the divisor by this term and subtract the result from the dividend to get a new polynomial.

4. Repeat the process with the new polynomial until the degree of the remainder is less than 2.

Try solving on your own before revealing the answer!

Final Answer: $r(x) = 4x - 3$

After performing the division, the remainder is $4x - 3$, which has degree 1 (less than 2).

Q2. Evaluate $\int_0^{\infty} \frac{3x^2}{(x^3 + 1)^3} dx$.

Background

Topic: Improper Integrals & Substitution

This question tests your ability to evaluate an improper integral using substitution and recognizing convergence.

Key Terms and Formulas

• Improper Integral: An integral with an infinite limit or an integrand with an infinite discontinuity.

• Substitution: Let $u = x^3 + 1$.

• Remember: $du = 3x^2 dx$.

Step-by-Step Guidance

1. Let $u = x^3 + 1$, so $du = 3x^2 dx$.

2. Change the limits of integration: When $x = 0$, $u = 1$; when $x \to \infty$, $u \to \infty$.

3. Rewrite the integral in terms of $u$ and simplify.

4. Set up the new integral and consider whether it converges.

Try solving on your own before revealing the answer!

Final Answer: $\frac{1}{2}$

After substitution and integration, the value is $\frac{1}{2}$, indicating the integral converges.

Q3. Which of the following statements about the integrals (i) $\int_{-1}^1 \frac{1}{x^2} dx$ and (ii) $\int_0^1 \frac{1}{e^{-x} + e^x} dx$ are correct?

Background

Topic: Improper Integrals & Convergence

This question tests your understanding of when an integral is improper and whether it converges or diverges.

Key Terms and Concepts

• Improper Integral: Has an infinite discontinuity or infinite limits.

• Convergent: The integral approaches a finite value.

• Divergent: The integral does not approach a finite value.

Step-by-Step Guidance

1. For (i): Check if $\frac{1}{x^2}$ is defined and continuous on $[-1, 1]$.

2. For (i): Identify if there is a discontinuity at $x = 0$ and determine if the integral is improper.

3. For (ii): Check if $\frac{1}{e^{-x} + e^x}$ is defined and continuous on $[0, 1]$.

4. For (ii): Determine if there are any discontinuities or infinite limits.

Try solving on your own before revealing the answer!

Final Answer: (B) (i) is not an improper integral; (ii) is a convergent improper integral.

Integral (i) is improper at $x=0$ and diverges; integral (ii) is proper and converges.

Q4. Which triangle is most suitable for setting up a trigonometric substitution for $\int \sqrt{x(x-2)} dx$ in terms of $\theta$?

Background

Topic: Trigonometric Substitution

This question tests your ability to recognize the appropriate right triangle for a trigonometric substitution involving square roots of quadratic expressions.

Key Terms and Formulas

• Trigonometric Substitution: Used for integrals involving $\sqrt{ax^2 + bx + c}$.

• Common substitutions: $x = a \sin \theta$, $x = a \cos \theta$, $x = a \tan \theta$.

Step-by-Step Guidance

1. Rewrite $\sqrt{x(x-2)}$ as $\sqrt{x^2 - 2x}$ if possible, or factor to match a standard form.

2. Identify which trigonometric identity (e.g., $1 - \sin^2 \theta = \cos^2 \theta$) would simplify the expression under the square root.

3. Match the expression to a triangle with sides corresponding to $x$, $x-2$, and the hypotenuse.

Try solving on your own before revealing the answer!

Final Answer: (B) Triangle with sides $x$, $x-2$, and angle $\theta$

This triangle setup matches the substitution needed for the integral.

Q5. Which is the correct form of the partial fraction decomposition of $\frac{1}{x^5 + 4x^3 + 4x}$?

Background

Topic: Partial Fraction Decomposition

This question tests your ability to factor a polynomial and write its rational function as a sum of simpler fractions.

Key Terms and Formulas

• Partial Fractions: Expressing a rational function as a sum of simpler rational expressions.

• Factorization: $x^5 + 4x^3 + 4x = x(x^4 + 4x^2 + 4)$

• Look for repeated and irreducible quadratic factors.

Step-by-Step Guidance

1. Factor the denominator completely.

2. Identify the types of terms (linear, repeated, irreducible quadratic) in the factorization.

3. Write the general form for the partial fraction decomposition based on the factors.

Try solving on your own before revealing the answer!

Final Answer: (A) $\frac{A}{x} + \frac{B}{x+2} + \frac{C}{(x+2)^2} + \frac{D}{x-2} + \frac{E}{(x-2)^2}$

This form matches the factorization and accounts for repeated linear factors.

Q6. Evaluate $\int_0^1 x^3 \sqrt{1-x^2} dx$.

Background

Topic: Integration by Substitution (Trigonometric Substitution)

This question tests your ability to use substitution, possibly trigonometric, to evaluate an integral involving a square root.

Key Terms and Formulas

• Let $x = \sin \theta$, so $dx = \cos \theta d\theta$.

• Limits: When $x = 0$, $\theta = 0$; when $x = 1$, $\theta = \frac{\pi}{2}$.

• $\sqrt{1-x^2} = \cos \theta$.

Step-by-Step Guidance

1. Let $x = \sin \theta$ and rewrite the integral in terms of $\theta$.

2. Substitute all $x$ and $dx$ terms and simplify the integrand.

3. Set up the new limits of integration and the simplified integral.

Try solving on your own before revealing the answer!

Final Answer: $\frac{2}{15}$

After substitution and integration, the value is $\frac{2}{15}$.

Q7. Evaluate $\int_{-1}^2 \frac{3x-4}{(x-3)(x+2)} dx$.

Background

Topic: Partial Fractions & Definite Integrals

This question tests your ability to decompose a rational function and integrate using partial fractions.

Key Terms and Formulas

• Partial Fraction Decomposition: $\frac{3x-4}{(x-3)(x+2)} = \frac{A}{x-3} + \frac{B}{x+2}$

• Integrate each term separately.

Step-by-Step Guidance

1. Decompose $\frac{3x-4}{(x-3)(x+2)}$ into partial fractions.

2. Integrate each term from $x = -1$ to $x = 2$.

3. Evaluate the resulting logarithmic expressions at the bounds.

Try solving on your own before revealing the answer!

Final Answer: $\ln(3) - \ln(2)$

The definite integral simplifies to this logarithmic expression.

Q8. Find the value of $A$ if $\int_0^{\pi/4} x \sec^2 x dx = A - \int_0^{\pi/4} \tan x dx$.

Background

Topic: Integration by Parts

This question tests your ability to use integration by parts and solve for a constant in an equation involving definite integrals.

Key Terms and Formulas

• Integration by Parts: $\int u dv = uv - \int v du$

• Let $u = x$, $dv = \sec^2 x dx$

• $\int \sec^2 x dx = \tan x$

Step-by-Step Guidance

1. Apply integration by parts to $\int x \sec^2 x dx$.

2. Evaluate the resulting expression from $0$to$\frac{\pi}{4}$.

3. Set up the equation and solve for $A$.

Try solving on your own before revealing the answer!

Final Answer: $A = 1$

After applying integration by parts and evaluating, $A$ is found to be 1.

Q9. Evaluate $\int x e^x dx$.

Background

Topic: Integration by Parts

This question tests your ability to use integration by parts for a product of a polynomial and an exponential function.

Key Terms and Formulas

• Integration by Parts: $\int u dv = uv - \int v du$

• Let $u = x$, $dv = e^x dx$

• $\int e^x dx = e^x$

Step-by-Step Guidance

1. Let $u = x$ and $dv = e^x dx$; compute $du$ and $v$.

2. Apply the integration by parts formula.

3. Simplify the resulting expression.

Try solving on your own before revealing the answer!

Final Answer: $x e^x - e^x + C$

This is the standard result for integrating $x e^x$.

Q10. Evaluate $\int_0^1 \frac{1}{x^2 - x} dx$.

Background

Topic: Improper Integrals & Partial Fractions

This question tests your ability to decompose a rational function and recognize improper integrals due to discontinuities in the interval.

Key Terms and Formulas

• Partial Fractions: $\frac{1}{x^2 - x} = \frac{A}{x} + \frac{B}{x-1}$

• Improper Integral: Discontinuity at $x = 0$ and $x = 1$.

Step-by-Step Guidance

1. Decompose $\frac{1}{x^2 - x}$ into partial fractions.

2. Set up the integral as the sum of two simpler integrals.

3. Check for discontinuities at the endpoints and determine if the integral converges.

Try solving on your own before revealing the answer!

Final Answer: The integral is divergent.

Because of the discontinuities at $x = 0$ and $x = 1$, the integral does not converge.

Q11. Evaluate $\int_0^1 \frac{1}{(x^2 + 1)^2} dx$.

Background

Topic: Integration Techniques (Trigonometric Substitution)

This question tests your ability to use trigonometric substitution or reduction formulas for integrals involving $(x^2 + 1)^2$.

Key Terms and Formulas

• Let $x = \tan \theta$, $dx = \sec^2 \theta d\theta$.

• Limits: When $x = 0$, $\theta = 0$; when $x = 1$, $\theta = \frac{\pi}{4}$.

• Use the reduction formula for $\int \sec^n \theta d\theta$ if needed.

Step-by-Step Guidance

1. Let $x = \tan \theta$ and rewrite the integral in terms of $\theta$.

2. Simplify the integrand and set up the new limits.

3. Integrate using the appropriate trigonometric identities or reduction formulas.

Try solving on your own before revealing the answer!

Final Answer: $\frac{\pi}{8} + \frac{1}{4}$

This is the result after substitution and integration.

Q12. Evaluate $\int_0^{\pi/2} \cos^3 x dx$.

Background

Topic: Integration of Powers of Trigonometric Functions

This question tests your ability to use reduction formulas or identities to integrate powers of cosine.

Key Terms and Formulas

• Reduction Formula: $\cos^n x = \cos^{n-2} x (1 - \sin^2 x)$

• Use the identity $\cos^3 x = \cos x (1 - \sin^2 x)$

• Integrate term by term.

Step-by-Step Guidance

1. Rewrite $\cos^3 x$ as $\cos x - \cos x \sin^2 x$.

2. Integrate each term separately over $[0, \frac{\pi}{2}]$.

3. Evaluate the definite integrals for each term.

Try solving on your own before revealing the answer!

Final Answer: $\frac{2}{3}$

This is the result after applying the reduction formula and evaluating the definite integral.