Mastering Integration by Parts

What is the integral of x * e^x dx using integration by parts?

• x e^x - e^x + C

• x^2 e^x + C

• e^x + C

• x e^x + e^x + C

Identify u and dv for the integral of ln(x) dx.

• u = 1, dv = ln(x) dx

• u = x, dv = ln(x) dx

• u = x^2, dv = dx

• u = ln(x), dv = dx

Evaluate the definite integral from 0 to 1 of x * sin(x) dx using integration by parts.

• sin(1) - cos(1)

• sin(1) - sin(0)

• sin(1) + cos(1)

• sin(0) - cos(0)

A company’s profit function is given by P(x) = x * e^(-x). Find the total profit from 0 to 5 units sold.

• 2.345

• 0.987

• 1.200

• 1.593

Determine u and dv for the integral of x^3 * ln(x) dx.

• u = x^3, dv = ln(x) dx

• u = x, dv = x^2 ln(x) dx

• u = ln(x^2), dv = x^3 dx

• u = ln(x), dv = x^3 dx

Evaluate the definite integral from 1 to 2 of x * ln(x) dx using integration by parts.

• 2 ln(2) - 3/4

• ln(2) - 1/2

• 2 - 3/4 ln(2)

• 3/4 ln(2) + 1

A car's distance traveled is modeled by the function s(t) = t^2 * e^(-t). Find the distance traveled from t=0 to t=3.

• 1.5

• 3.5

• 0.5

• 2.0

Use integration by parts to find the integral of x * tan(x) dx.

• x * ln|sin(x)| + C

• -x * cos(x) + C

• x^2 * tan(x) + C

• -x * ln|cos(x)| + x + C

Identify u and dv for the integral of e^(2x) * sin(3x) dx.

• u = e^(2x), dv = sin(3x) dx

• u = sin(3x), dv = e^(2x) dx

• u = sin(3x), dv = e^(3x) dx

• u = cos(3x), dv = e^(2x) dx

Evaluate the definite integral from 0 to π/2 of x * cos(x) dx using integration by parts.

• (π/2) + 2

• (π/2) - 2

• (π/2) + 1

• (π/2) - 1