INTEGRATION TECHNIQUES

Multiple Choice

What are some techniques for finding antiderivatives of functions?

• Using the power rule

• Using integration by parts

• Using substitution

• All of the above

Multiple Choice

What is the integral of 2x cos(x²) dx?

• sin(x²) + C

• cos(x²) + C

• tan(x²) + C

• sec(x²) + C

Multiple Choice

What is the integral of the function represented in the equation?

• (1/5)(1-x^2) - (1/3)

• (1/2)(1-u)√u du

• (1/2)(1-x^2)

• (1/3)(1-x^2)

Multiple Choice

Evaluate \( \int_2^4 x sin(x^2) dx \). What substitution should be made to solve this integral?

• Let \( u = x^2 \)

• Let \( u = 2x \)

• Let \( u = sin(x) \)

• Let \( u = cos(x) \)

Open Ended

Evaluate \( \int_{1/4}^{1/2} \frac{\cos(\pi t)}{\sin^2(\pi t)} dt \) and explain the steps involved in the evaluation.

Multiple Choice

Evaluate \( \int sin^5 x \: dx \)

• \( \int sin^4 x \: dx \)

• \( \int sin^2 x \: dx \)

• \( \int sin x(1 - cos^2 x)^2 \: dx \)

• \( \int sin x \: dx \)

Open Ended

Evaluate \( \int \sin^2 x \cos^2 x \ dx \). Use the formulas \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) and \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) to get: \( \int \sin^2 x \cos^2 x \ dx = \int \frac{1 - \cos(2x)}{2} \cdot \frac{1 + \cos(2x)}{2} \ dx \).

Multiple Choice

Evaluate the integral \( \int \sqrt{1-x^2} \: dx \) using the substitution \( x = sin u \). What is the resulting expression after substitution?

• \( \frac{arc sin x}{2} + \frac{2x \sqrt{1-x^2}}{4} + C \)

• \( \frac{arc sin x}{2} + \frac{x \sqrt{1-x^2}}{2} + C \)

• \( \frac{arc sin x}{2} + \frac{2 \sqrt{1-x^2}}{4} + C \)

• \( \frac{arc sin x}{2} + \frac{2x \sqrt{1-x^2}}{2} + C \)

Multiple Choice

This can be integrated using u-substitution. What should "u" equal in this integral?

• x

• (lnx)/x

• lnx

• sin(lnx)

Multiple Choice

What is the integral of sec u du?

• sec u + C

• ln |sec u + tan u| + C

• tan u + C

• sec^2 u + C

Multiple Choice

What is the integral of the function \( \arctan(x) \)?

• \( x \arctan(x) - \frac{1}{2} \ln(1+x^2) + C \)

• \( \ln(x) + C \)

• \( x^2 + C \)

• \( \arctan(x) + C \)

Multiple Choice

Evaluate the integral \( \int \frac{x^3}{(x-2)(x+3)} dx \) as the sum of two fractions.

• \( \frac{7x-6}{(x-2)(x+3)} = \frac{A}{x-2} + \frac{B}{x+3} \)

• \( \frac{7x-6}{(x-2)(x+3)} = \frac{A}{x+3} + \frac{B}{x-2} \)

• \( \frac{7x-6}{(x-2)(x+3)} = \frac{A}{x-2} + \frac{B}{x-3} \)

• \( \frac{7x-6}{(x-2)(x+3)} = \frac{A}{x+2} + \frac{B}{x+3} \)

Open Ended

Find the antiderivatives of the following functions: 1. ∫ 1/(4 - x²) dx 2. ∫ x⁴/(4 - x²) dx 3. ∫ x²/(4 - x²) dx 4. ∫ 1/(x² + 10x + 25) dx 5. ∫ x⁴/(4 + x²) dx 6. ∫ 1/(x² + 10x + 29) dx 7. ∫ x³/(4 + x²) dx 8. ∫ 1/(x² + 10x + 21) dx 9. ∫ 1/(2x² - x - 3) dx 10. ∫ 1/(x² + 3x) dx

Multiple Choice

Which of the following is a correct formula for the Trapezoidal Rule for approximating $$\int_{a}^{b} f(x) dx$$ with $$n$$ subintervals?

• $$\frac{b-a}{n} \left[ \frac{f(a) + f(b)}{2} + \sum_{k=1}^{n-1} f(a + k\frac{b-a}{n}) \right]$$

• $$\frac{b-a}{2n} \left[ f(a) + 2\sum_{k=1}^{n-1} f(a + k\frac{b-a}{n}) + f(b) \right]$$

• $$\frac{b-a}{3n} \left[ f(a) + 4\sum_{k=1}^{n-1} f(a + k\frac{b-a}{n}) + f(b) \right]$$

• $$\frac{b-a}{n} \sum_{k=0}^{n} f(a + k\frac{b-a}{n})$$

Multiple Choice

What is the trapezoid approximation for the integral \( \int_0^1 e^{-x^2} dx \) using six intervals?

• 0.74982

• 0.74042

• 0.74512

• 0.7727

Multiple Choice

What is the formula for the area under the parabola when approximating the integral?

• \( \frac{\Delta x}{3} (f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + f(x_n)) \)

• \( \frac{\Delta x}{2} (f(x_0) + f(x_n)) \)

• \( \Delta x (f(x_0) + f(x_n)) \)

• \( \frac{\Delta x}{4} (f(x_0) + 2f(x_1) + 2f(x_2) + f(x_n)) \)

Multiple Choice

What is the error estimate formula for Simpson's approximation as given in the theorem?

• E(Δx) = (b - a)M(Δx)^4 / 180n^4

• E(Δx) = (b - a)M(Δx)^5 / 180n^4

• E(Δx) = (b - a)M(Δx)^3 / 180n^4

• E(Δx) = (b - a)M(Δx)^2 / 180n^4

Multiple Choice

Using Simpson’s rule on a parabola f(x), even with just two subintervals, gives the exact value for the integral. What is the formula for f(x) in terms of x?

• f(x) = x^2 - x

• f(x) = 3x^2 + 4x

• f(x) = ax^3 + bx^2 + cx + d

• f(x) = 2x^2 + 5x

Multiple Choice

What would you choose for your u here if you used integration by parts?

• ln x

• ln (x) sin (x)

• sin (x)

• x

Poll

How confident do you feel about this topic now?

• Very confident
• Somewhat confident
• Not confident