403 results for "integration".

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• Exam (13 questions)

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

• Question

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

• Question

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

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Find $\displaystyle\int \frac{ax^3-ax+b}{1-x^2}\;dx$. Input constant of integration as $C$.

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

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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

Find $\displaystyle \int ax ^ m+ bx^{c/n}\;dx$.

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Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$.

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Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

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

• Exam (8 questions)

Find an integral by choosing a suitable substitution.

• Question

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$

• Exam (4 questions)

Find the integral of an improper fraction.

• Exam (11 questions)

Questions which rely on knowledge of standard integrals.

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Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

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Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

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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

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

Find $\displaystyle \int (ax+b)\cos(cx+d)\; dx$ and hence find $\displaystyle \int (ax+b)^2\sin(cx+d)\; dx$

• 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

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

• Exam (3 questions)

Integrate various functions by rewriting them as partial fractions.

• Exam (6 questions)

Integrate the product of two functions by the method of integration by parts.

• 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.

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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.

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

• Question

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 $g(x)$.

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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

Integrating by parts.

Find $\int ax\sin(bx+c)\;dx$ or $\int ax e^{bx+c}\;dx$

• Question

$f(x,y)$ 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 $P(Y \le b|X \ge c)$