Mastering Integration Techniques

What is the integral of x^n with respect to x?

• (x^(n+1))/(n+1) + C

• (x^(n-1))/(n-1) + C

• (n*x^(n+1))/2 + C

• (x^(n+2))/(n+2) + C

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

• 45

• 72

• 63

• 50

Find the indefinite integral of e^x.

• e^x + C

• e^x + 1

• e^x - C

• e^(x+1) + C

Use integration by parts to evaluate ∫ x * sin(x) dx.

• -x * sin(x) + cos(x) + C

• x * sin(x) - cos(x) + C

• x * cos(x) + sin(x) + C

• -x * cos(x) + sin(x) + C

What is the substitution method in integration?

• The substitution method is a technique used in integration to simplify integrals by changing the variable of integration.

• A method to find the derivative of a function.

• A way to approximate the value of an integral.

• A technique for solving differential equations.

Evaluate ∫ (2x + 1)^5 dx using substitution.

• (1/6)(2x + 1)^5 + C

• (1/12)(2x + 1)^6 + C

• (1/10)(2x + 1)^6 + C

• (1/8)(2x + 1)^7 + C

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

• 4

• 8

• 2

• 6

Calculate the integral of cos(2x) dx.

• (1/2) sin(2x) + C

• sin(2x) + C

• (1/2) cos(2x) + C

• (1/4) sin(4x) + C

Use integration by parts to solve ∫ ln(x) dx.

• x ln(x) + x + C

• ln(x^2)/2 + C

• x ln(x) - x + C

• x^2/2 + C

Evaluate the definite integral ∫_0^1 (4 - x^2) dx.

• 2/3

• 11/3

• 4

• 5/3