Exam questions on integration?

1. Calculate the definite integral of f(x) = 3x^2 + 2x + 1 from x = 0 to x = 2.
2. Find the indefinite integral of g(x) = 4x^3 + 5x^2 - 2x + 3.
3. Use integration by parts to evaluate the integral of h(x) = x*sin(x) dx.
4. Find the area enclosed by the curve y = x^2 and the x-axis from x = 0 to x = 3.
5. Evaluate the integral of the function k(x) = e^x / (1 + e^x) dx.
6. Use substitution to evaluate the integral of l(x) = 2x*cos(x^2) dx.
7. Find the volume of the solid generated by revolving the region bounded by y = x^2, y = 0, and x = 2 about the y-axis.
8. Evaluate the integral of m(x) = 1 / (x^2 + 1) dx using partial fractions.
9. Determine the average value of the function n(x) = 2x^2 + 3x - 1 on the interval [0, 4].
10. Use trigonometric substitution to evaluate the integral of p(x) = sqrt(9 - x^2) dx.

Basic Integration

- Evaluate the integral: ∫x^2 dx
- Find the area under the curve y = f(x) over the interval [a, b].

Integration by Substitution

- Use u-substitution to evaluate the integral: ∫sin(x)cos(x) dx
- Find the integral: ∫√(x^2 + 1) dx

Integration by Parts

- Use integration by parts to evaluate the integral: ∫x e^x dx
- Find the integral: ∫ln(x) dx

Trigonometric Integrals

- Evaluate the integral: ∫sin(2x) dx
- Find the integral: ∫cos^2(x) dx

Improper Integrals

- Determine if the integral ∫1/(x^2) dx from 0 to 1 converges or diverges.
- Evaluate the improper integral: ∫e^(-x^2) dx from -∞ to ∞

Applications of Integration

- Find the volume of the solid generated by rotating the region under the curve y = f(x) over the interval [a, b] about the x-axis.
- Calculate the work done by a force over a distance.

Advanced Integration Techniques

- Use partial fractions to evaluate the integral: ∫(x^2 + 2x - 3)/(x^3 - 9x) dx
- Find the integral: ∫e^(x^2) dx
- Evaluate the double integral: ∫∫(x + y) dA over the rectangular region [0, 2] × [0, 1]