Mathematics

1. 1.

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

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Integrate term by term using the standard results for x^n, 1/x and e^(kx).
   2. For e^(kx) divide by k; for 1/x use the natural logarithm of the modulus.

   Marking points

   • Integrates 6x^2 to 2x^3.
   • Integrates -4/x to -4 ln|x|.
   • Integrates e^(2x) to (1/2)e^(2x) and adds the constant: 2x^3 - 4 ln|x| + (1/2)e^(2x) + C.

   Examiner tip: Include + C for an indefinite integral and divide by k when integrating e^(kx).

2. 2.

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

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The derivative of cos(2x) is -2 sin(2x), so divide by 2 and keep the minus sign.
   2. Work in radians and subtract lower from upper.

   Marking points

   • An antiderivative is -(1/2)cos(2x).
   • Substitutes the limits: -(1/2)cos(pi) + (1/2)cos(0).
   • The value is 1/2 + 1/2 = 1.

   Examiner tip: Use radians, and take care with the minus sign when integrating sine.

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

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The factor x dx is exactly half of du, so the substitution removes x completely.
   2. Convert the limits to u so there is no need to change back to x.

   Marking points

   • du = 2x dx, so x dx = (1/2) du.
   • Changes the limits: x = 0 gives u = 1 and x = sqrt(3) gives u = 4.
   • Integrates (1/2)u^(1/2) to (1/3)u^(3/2).
   • Evaluates (1/3)(8 - 1) = 7/3.

   Examiner tip: Change the limits when you change the variable, or return to x before substituting limits. Never mix the two.

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

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Choose u = ln x because its derivative 1/x simplifies the remaining integral, and ln x cannot be integrated directly.
   2. Apply the parts formula, then evaluate with the limits 1 and e (ln 1 = 0, ln e = 1).

   Marking points

   • Chooses u = ln x and dv = x dx so du = 1/x dx and v = x^2/2.
   • Writes [x^2 ln x / 2] - integral of x/2 dx.
   • Evaluates the first part as e^2/2 and the integral as (e^2 - 1)/4.
   • The result is e^2/2 - (e^2 - 1)/4 = (e^2 + 1)/4.

   Examiner tip: When the integrand contains ln x, let u = ln x. Check by differentiating your answer.

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

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Split the rational function into simple fractions that each integrate to a logarithm.
   2. After substituting limits, use n ln a = ln(a^n) and ln a - ln b = ln(a/b) to reach a single logarithm.

   Marking points

   • Writes the integrand as 2/(x - 1) + 3/(x + 2).
   • Integrates to 2 ln(x - 1) + 3 ln(x + 2).
   • Substitutes the limits: (2 ln 2 + 3 ln 5) - (0 + 3 ln 4).
   • Simplifies 3 ln 4 = 6 ln 2 to get 3 ln 5 - 4 ln 2.
   • Combines using logarithm laws: ln(5^3/2^4) = ln(125/16).

   Examiner tip: In a 'show that' question, write every step. Convert 4 as 2^2 so the logarithms can be combined.

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]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The line y = 2x lies above the parabola between the intersections, so subtract curve from line for area.
   2. For the volume, the solid is a disc with a hole: subtract the inner volume from the outer volume, squaring each y before integrating.

   Marking points

   • Finds the intersections: x^2 = 2x gives x = 0 and x = 2.
   • Area = integral of (2x - x^2) dx from 0 to 2.
   • Evaluates the area as 4 - 8/3 = 4/3.
   • Volume = pi * integral of ((2x)^2 - (x^2)^2) dx, a difference of squares.
   • Integrates 4x^2 - x^4 to 4x^3/3 - x^5/5.
   • Volume = pi(32/3 - 32/5) = 64 pi/15.

   Examiner tip: For a region between two curves rotated about the x-axis, subtract the squares, not the square of the difference.

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