Integrales

What is the difference between definite and indefinite integrals?

• Indefinite integrals have specific limits of integration

• The main difference is that definite integrals have specific limits of integration, while indefinite integrals do not have limits and include a constant of integration.

• Definite integrals do not include a constant of integration

• Definite integrals have no limits of integration

Explain the concept of integration by parts and provide an example.

• To integrate by parts, we choose one function as u and the other as dv, then differentiate u and integrate dv to apply the formula. For example, ∫x*sin(x) dx can be solved using integration by parts by choosing u = x and dv = sin(x) dx.

• Integration by multiplication requires dividing the functions instead of applying the formula

• Integration by subtraction involves adding two functions instead of multiplying them

• Integration by division involves taking the derivative of one function and integrating the other

How does integration by substitution work? Give an example to illustrate.

• For example, consider the integral ∫2x * cos(x^2) dx. Let u = x^2, then du = 2x dx. Substituting u and du into the integral gives ∫cos(u) du, which is easier to solve. After finding the antiderivative, the final answer is sin(u) + C, where C is the constant of integration. Finally, substituting back x^2 for u gives sin(x^2) + C as the solution.

• The substitution method can only be applied to definite integrals.

• Integration by substitution involves multiplying the integrand by a constant before substitution.

• For example, consider the integral ∫2x * sin(x^2) dx. Let u = x^2, then du = 2x dx. Substituting u and du into the integral gives ∫cos(u) du, which is easier to solve. After finding the antiderivative, the final answer is sin(u) + C, where C is the constant of integration. Finally, substituting back x^2 for u gives sin(x^2) + C as the solution.

List some common integration techniques used in calculus.

• inverse trigonometric integrals

• substitution, integration by parts, trigonometric integrals, partial fractions, trigonometric substitution

• differentiation by parts

• exponential integrals

What are some applications of integrals in real-life scenarios?

• Some applications of integrals in real-life scenarios include calculating areas under curves, determining volumes of irregular shapes, finding center of mass, calculating work done by a force, and analyzing population growth.

• Utilizing integrals to design clothing patterns

• Applying integrals to predict the weather

• Using integrals to bake a cake

Calculate the definite integral of f(x) = 2x from 0 to 3.

• 9

• 5

• 12

• 6

Find the indefinite integral of g(x) = 3x^2 + 2x - 5.

• 3x^2 + 2x + 5

• x^3 + x^2 - 5

• x^3 + x^2 - 5x + C

• 3x^3 + 2x^2 - 5x

Solve the integral ∫(x^2 + 2x) dx using integration by parts.

• 2x^2 + C

• (1/2)x^3 + C

• (1/3)x^3 + C

• x^2 + C

Use integration by substitution to evaluate the integral ∫(2x + 1)^2 dx.

• (1/4) * (2x + 1)^3 + C

• (1/3) * (2x + 1)^3 + C

• (1/6) * (2x + 1)^3 + C

• (1/5) * (2x + 1)^3 + C

Apply the appropriate integration technique to solve the integral ∫(cos(x) + 2sin(x)) dx.

• cos(x) - 2sin(x) + C

• cos(x) + sin(x) + C

• sin(x) - 2cos(x) + C

• sin(x) + 2cos(x) + C