403 results for "integration".

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

Question in Content created by Newcastle University
Multiple response question (2 correct out of 4) covering properties of Riemann integration. Selection of questions from a pool.

Question in Content created by Newcastle University
$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1n}+C$ for a polynomial $g(x)$. Find $g(x)$.

Question in Content created by Newcastle University
Find $\displaystyle\int \frac{ax^3ax+b}{1x^2}\;dx$. Input constant of integration as $C$.

Question in Content created by Newcastle University
Integrate $f(x) = ae ^ {bx} + c\sin(dx) + px^q$. Must input $C$ as the constant of integration.

Question in Content created by Newcastle University
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.

Question in Content created by Newcastle University
Find $\displaystyle \int ax ^ m+ bx^{c/n}\;dx$.

Question in Content created by Newcastle University
Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$.

Question in Content created by Newcastle University
Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

Question in Content created by Newcastle University
Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$

Exam (8 questions) in Content created by Newcastle University
Find an integral by choosing a suitable substitution.

Question in Content created by Newcastle University
Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$

Exam (4 questions) in Content created by Newcastle University
Find the integral of an improper fraction.

Exam (11 questions) in Content created by Newcastle University
Questions which rely on knowledge of standard integrals.

Question in Content created by Newcastle University
Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

Question in Content created by Newcastle University
Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

Question in Content created by Newcastle University
Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$.

Question in Content created by Newcastle University
Calculate definite integrals: $\int_0^\infty\;e^{ax}\,dx$, $\int_1^2\;\frac{1}{x^{b}}\,dx$, $\; \int_0^{\pi}\;\cos\left(\frac{x}{2n}\right)\,dx$

Question in Content created by Newcastle University
Find $\displaystyle \int (ax+b)\cos(cx+d)\; dx $ and hence find $\displaystyle \int (ax+b)^2\sin(cx+d)\; dx $

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

Exam (3 questions) in Content created by Newcastle University
Integrate various functions by rewriting them as partial fractions.

Exam (6 questions) in Content created by Newcastle University
Integrate the product of two functions by the method of integration by parts.

Question in Content created by Newcastle University
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 Content created by Newcastle University
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 Content created by Newcastle University
Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

Question in Content created by Newcastle University
Given that $\displaystyle \int x({ax+b)^{m}} dx=\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.

Question in Content created by Newcastle University
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 Content created by Newcastle University
Integrating by parts.
Find $ \int ax\sin(bx+c)\;dx$ or $\int ax e^{bx+c}\;dx$

Question in Content created by Newcastle University
$f(x,y)$ is the PDF of a bivariate distribution $(X,Y)$ on a given rectangular region in $\mathbb{R}^2$. Write down the limits of the integrations needed to find $P(X \ge a)$, the marginal distributions $f_X(x),\;f_Y(y)$ and the conditional probability $P(Y \le bX \ge c)$