$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$  for a polynomial $g(x)$. Find $g(x)$.

Find $\displaystyle I=\int \frac{2 a x + b} {a x ^ 2 + b x + c}\;dx$ by substitution or otherwise.

Evaluate $\int_0^{\,m}e^{ax}\;dx$, $\int_0^{p}\frac{1}{bx+d}\;dx,\;\int_0^{\pi/2} \sin(qx) \;dx$. 

No solutions given in Advice to parts a and c.

Tolerance of 0.001 in answers to parts a and b. Perhaps should indicate to the student that a tolerance is set. The feedback on submitting an incorrect answer within the tolerance says that the answer is correct - perhaps there should be a different feedback in this case if possible for all such questions with tolerances.

Multiple response question (2 correct out of 4) covering properties of Riemann integration. Selection of questions from a pool.

Equations which can be written in the form

\[\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x),   \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(y),   \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x)f(y)\]

can all be solved by integration.

In each case it is possible to separate the $x$'s to one side of the equation and the $y$'s to the other

Solving such equations is therefore known as solution by separation of variables

Straightforward question: student must find the general solution to a second order constant coefficient ODE. Uses custom marking algorithm to check that both roots appear and that the solution is in the correct form (e.g. two arbitrary constants are present). Arbitrary constants can be any non space-separated string of characters. The algorithm also allows for the use of $e^x$ rather than $\exp(x)$.

Unit tests are also included, to check whether the algorithm accurately marks when the solution is correct; when it's correct but deviates from the 'answer'; when one or more roots is incorrect; or when the roots are correct but constants of integration have been forgotten.

Find roots and the area under a parabola

Find roots and the area under a parabola

Find roots and the area under a parabola

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Basic indefinite integrals, Basic definite integrals, integration by substitution

Missing: Area type question, solving diff eq application

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Straightforward question: student must find the general solution to a second order constant coefficient ODE. Uses custom marking algorithm to check that both roots appear and that the solution is in the correct form (e.g. two arbitrary constants are present). Arbitrary constants can be any non space-separated string of characters. The algorithm also allows for the use of $e^x$ rather than $\exp(x)$.

Unit tests are also included, to check whether the algorithm accurately marks when the solution is correct; when it's correct but deviates from the 'answer'; when one or more roots is incorrect; or when the roots are correct but constants of integration have been forgotten.

Find roots and the area under a parabola

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Three graphs are given with areas underneath them shaded. The student is asked to calculate their areas, using integration.  Q1 has a polynomial. Q2 has exponentials and fractional functions. Q3 requires integration by parts.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

This quiz is designed to help you practise your integration techniques.

Improvements needed:

• Add some definite integrals to Q2
• Use different variables

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

4 questions on using partial fractions to solve indefinite integrals.

4 questions on integrating by parts.

4 questions on using partial fractions to solve indefinite integrals.

4 questions on integrating by parts.

5 questions on using substitution to find indefinite integrals.

5 questions on using substitution to find indefinite integrals.

5 questions on indefinite integration. Includes integration by parts and integration by substitution.

5 questions on indefinite integration. Includes integration by parts and integration by substitution.