Mathematics

1. 1.

   Find the integral of (6x^2 - 4/x + e^(2x)) with respect to x.

   [3 marks] · no calculator
2. 2.

   Evaluate the definite integral of sin(2x) from x = 0 to x = pi/2.

   [3 marks] · no calculator
3. 3.

   Use the substitution u = x^2 + 1 to evaluate the integral of x sqrt(x^2 + 1) from x = 0 to x = sqrt(3).

   [4 marks] · no calculator
4. 4.

   Use integration by parts to find the exact value of the integral of x ln x from x = 1 to x = e.

   [4 marks] · no calculator
5. 5.

   Using partial fractions, show that the integral of (5x + 1)/((x - 1)(x + 2)) from x = 2 to x = 3 equals ln(125/16).

   [5 marks] · no calculator
6. 6.

   The region R is bounded by the curves y = x^2 and y = 2x. Find (a) the area of R, (b) the exact volume when R is rotated through 360 degrees about the x-axis.

   [6 marks]

Original, unofficial practice questions from SubjectScout — not official past-paper questions.