403 results for "integration".

Question in Calculus Math 5A
Integration by susbtitution, no hint given

Question in Calculus Math 5A
Simple Indefinite Integrals

Exam (13 questions) in Sharika's workspace
Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Question in Christian's workspace
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 in Andrew's workspace
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 spaceseparated 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 in Shivram's workspace
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 in Shivendra's workspace
A graph is drawn. A student is to identify the derivative of this graph from four other graphs.
Version I. Graph is quadratic
Version II. Graph is horizontal
Version III. Graph is cubic
Version IV. Graph is sinusoidal

Exam (13 questions) in Kevin's workspace
Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Question in Kevin's workspace
Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$

Question in Kevin's workspace
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 in Kevin's workspace
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 in MY QUESTIONS
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 in MY QUESTIONS
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 in MY QUESTIONS
Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

Question in MY QUESTIONS
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 in MY QUESTIONS
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 in MY QUESTIONS
Find $\displaystyle \int (ax)\ln(cx)\; dx $

Question in MY QUESTIONS
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 in MY QUESTIONS
5 indefinite integrals containing exponential functions
rebelmaths

Question in MY QUESTIONS
Find roots and the area under a parabola

Question in MY QUESTIONS
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 xaxis.

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

Exam (11 questions) in Maria's workspace
Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Exam (3 questions) in Maria's workspace
Integrate various functions by rewriting them as partial fractions.

Exam (11 questions) in Maria's workspace
Questions which rely on knowledge of standard integrals.

Exam (4 questions) in Maria's workspace
Find the integral of an improper fraction.

Exam (4 questions) in Maria's workspace
4 questions on integrating by parts.