Mastering Integration Concepts

What is the value of the definite integral ∫_0^1 (x^2) dx?

• 2/3

• 1/3

• 1/2

• 1/4

Evaluate the definite integral ∫_1^2 (3x^2 - 2) dx.

• 4

• 5

• 7

• 6

What is the Fundamental Theorem of Calculus?

• The Fundamental Theorem of Calculus states that differentiation is the same as multiplication.

• The Fundamental Theorem of Calculus defines the area under a curve as a constant.

• The Fundamental Theorem of Calculus states that all functions are continuous.

• The Fundamental Theorem of Calculus states that integration and differentiation are inverse processes.

Explain the difference between definite and indefinite integrals.

• Indefinite integrals represent area under a curve, while definite integrals represent a constant value.

• Definite integrals yield a function, while indefinite integrals yield a number.

• Definite integrals are always negative, while indefinite integrals are always positive.

• Definite integrals yield a number representing area, while indefinite integrals yield a function plus a constant.

Use substitution to evaluate the integral ∫ (2x) * e^(x^2) dx.

• e^(x^2) + C

• e^(x^2) * ln(x) + C

• 2x * e^(x^2) + C

• x * e^(x^2) + C

What technique would you use to integrate ∫ (sin(x) * cos(x)) dx?

• (1/4)sin(2x) + C

• -(1/4)cos(2x) + C

• (1/2)sin(2x) + C

• (1/2)cos(2x) + C

Apply integration by parts to ∫ x * e^x dx.

• (x - 1)e^x + C

• (x + 1)e^x + C

• xe^x + C

• (x^2)e^x + C

What is the formula for integration by parts?

• ∫u dv = u^2/2 - ∫v du

• ∫u dv = uv + ∫v du

• ∫u dv = uv - ∫v du

• ∫u dv = uv + ∫u du

Evaluate the integral ∫ (ln(x)) dx using integration by parts.

• x ln(x) - x + C

• x^2/2 + C

• e^x + C

• x ln(x) + x + C

What is the result of the definite integral ∫_0^π (sin(x)) dx?

• π

• 0

• 1

• 2