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106 results.

Calculate line integral over a straight line
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find $\displaystyle \int_{\Gamma} \left(x+y \right)\;dx+\left(y-x\right)\;dy,\;\Gamma$ is the line from $(0,0)$ to $(a,b)$ .
Construct a probability distribution function, then find CDF and expectation
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

The random variable $X$ has a PDF which involves a parameter $c$ . Find the value of $c$ . Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$ .

Also find the expectation $\displaystyle \operatorname{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\;dx$ .
Integral of improper polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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

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

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

Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$ . Solution involves $\arctan$ .
Integration: Definite Integration
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

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.
Indefinite integral of improper polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle\int \frac{ax^3-ax+b}{1-x^2}\;dx$ . Input constant of integration as $C$ .
Integration: Indefinite integral by substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$
Indefinite integral: polynomial fraction
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find the polynomial $g(x)$ such that $\displaystyle \int \frac{ax+b}{(cx+d)^{n}} dx=\frac{g(x)}{(cx+d)^{n-1}}+C$ .
Integration by parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$ , find $g(x)$ . Find $h(x)$ , $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$ .
Integration of fraction with power in denominator
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$ for a polynomial $g(x)$ . Find $g(x)$ .
Calculate definite integral with a hyperbolic substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$
Indefinite integral of polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{nx^3+mx^2+px +m}{x^2+1} \;dx$
Definite integration using standard integrals
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Evaluate $\int_1^{\,m}(ax ^ 2 + b x + c)^2\;dx$ , $\int_0^{p}\frac{1}{x+d}\;dx,\;\int_0^\pi x \sin(qx) \;dx$ , $\int_0^{r}x ^ {2}e^{t x}\;dx$
Definite integration using standard integrals
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Calculate definite integrals: $\int_0^\infty\;e^{-ax}\,dx$ , $\int_1^2\;\frac{1}{x^{b}}\,dx$ , $\; \int_0^{\pi}\;\cos\left(\frac{x}{2n}\right)\,dx$
Indefinite integral of improper polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$ . Solution involves $\arctan$ .
Integral by hyperbolic substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$
Integration by parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$
Indefinite integral by substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$ . Solution involves the inverse trigonometric function $\arcsin$ .
Integration: Indefinite integral
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int ax ^ m+ bx^{c/n}\;dx$ .
Integration: Indefinite integral
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$ .
Integration: Indefinite integral
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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

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 $g(x)$ .
Integration by partial fractions
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d$ .
Find indefinite integrals of hyperbolic functions
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find $\displaystyle \int\cosh(ax+b)\;dx,\;\;\int x\sinh(cx+d)\;dx$ .

Advice tidied up.
Integration By Parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Integrating by parts.

Find $\int ax\sin(bx+c)\;dx$ or $\int ax e^{bx+c}\;dx$
Integration by partial fractions
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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.
Integration by partial fractions
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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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