403 results for "integration".

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• Question

Integration by susbtitution, no hint given

• Question

Simple Indefinite Integrals

• Exam (13 questions)

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

• Question

The student must perform an indefinite integration, and include a constant of integration.

A custom marking algorithm identifies the name of the student's constant of integration, and replaces it with 'c' before comparing with the expected answer.

If the student enters something like '2c' instead of 'c', it'll be marked wrong, which is an error in the marking.

• Question

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.

• Question in HNC Exam

Evaluate $\int_1^{\,m}(ax ^ 2 + b x + c)^2\;dx$, $\int_0^{p}\frac{1}{x+d}\;dx,\;\int_0^\pi x \sin(qx) \;dx$, $\int_0^{r}x ^ {2}e^{t x}\;dx$

• Question

Evaluate $\int_1^{\,m}(ax ^ 2 + b x + c)^2\;dx$, $\int_0^{p}\frac{1}{x+d}\;dx,\;\int_0^\pi x \sin(qx) \;dx$, $\int_0^{r}x ^ {2}e^{t x}\;dx$

• Question in HNC Exam

Evaluate $\int_1^{\,m}(ax ^ 2 + b x + c)^2\;dx$, $\int_0^{p}\frac{1}{x+d}\;dx,\;\int_0^\pi x \sin(qx) \;dx$, $\int_0^{r}x ^ {2}e^{t x}\;dx$

• Question

A graph is drawn. A student is to identify the derivative of this graph from four other graphs.

Version II. Graph is horizontal

Version III. Graph is cubic

Version IV. Graph is sinusoidal

• Integration
Draft
Exam (13 questions)

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

• Question

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$

• Question

Two quadratic graphs are sketched with some area beneath them shaded. Question is to determine the area of shaded regions using integration. The first graph's area is all above the $x$-axis. The second graph has some area above and some below the $x$-axis.

• Question

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx$ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$

• Question

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

• Question

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

• Question

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d$.

• Question

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.

• Question

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx$ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$

• Question

Find $\displaystyle \int (ax)\ln(cx)\; dx$

• Question

Two quadratic graphs are sketched with some area beneath them shaded. Question is to determine the area of shaded regions using integration. The first graph's area is all above the $x$-axis. The second graph has some area above and some below the $x$-axis.

• Question

5 indefinite integrals containing exponential functions

rebelmaths

• Question

Find roots and the area under a parabola

• Question

Function $f(x) = xe^{ax}$ is sketched and area shaded. Question is to determine the area under graph, exactly and (calculator) to 3 s.f. Area is above x-axis.

• Question

Question is to calculate the area bounded by a cubic and the $x$-axis. Requires finding the roots by solving a cubic equation. Calculator question

• Question

Graphs are given with areas underneath them shaded. The student is asked to select the correct integral which calculates its area.

• Exam (11 questions)

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

• Exam (3 questions)

Integrate various functions by rewriting them as partial fractions.

• Exam (11 questions)

Questions which rely on knowledge of standard integrals.

• Exam (4 questions)

Find the integral of an improper fraction.

• Exam (4 questions)

4 questions on integrating by parts.