Mastering Integrals

Evaluate the definite integral of f(x) = 3x^2 from x = 1 to x = 4.

• 63

• 72

• 50

• 45

Find the indefinite integral of g(x) = 5e^x.

• 5e^x + C

• 5e^x - C

• 5e^x + C

• 5e^x + 1

Use substitution to evaluate the integral of h(x) = sin(2x) dx.

• -(1/2)cos(2x) + C

• (1/2)sin(2x) + C

• -sin(2x) + C

• (1/2)cos(2x) + C

Calculate the area under the curve of y = x^3 from x = 0 to x = 2.

• 4

• 8

• 6

• 2

Solve the real-world problem: A tank is being filled at a rate of 4t liters per hour. How much water is added from t = 0 to t = 3 hours?

• 18 liters

• 6 liters

• 12 liters

• 24 liters

Evaluate the definite integral of f(x) = 2x + 1 from x = 0 to x = 5.

• 30

• 25

• 35

• 20

Find the indefinite integral of g(x) = 1/(x^2 + 1).

• e^x + C

• arctan(x) + C

• ln(x) + C

• sin(x) + C

Use substitution to evaluate the integral of h(x) = (3x^2)(e^(x^3)) dx.

• 3x^2 + C

• (e^(x^3))^2 + C

• e^(x^3) + C

• (x^3)(e^(x^3)) + C

Calculate the area under the curve of y = 2x from x = 1 to x = 3.

• 10

• 4

• 8

• 6

Solve the real-world problem: A car's velocity is given by v(t) = 3t^2 + 2. How far does it travel from t = 1 to t = 4?

• 69

• 50

• 72

• 85