Luis Hernandez's content - Numbas at mathcentre.ac.uk

216 results authored by Luis Hernandez - search across all users.

Luis's copy of Indefinite integral of polynomial fraction
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{nx^3+mx^2+px +m}{x^2+1} \;dx$
Luis's copy of Indefinite integral of improper polynomial fraction
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$ . Solution involves $\arctan$ .
Luis's copy of Indefinite integral of improper polynomial fraction
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle\int \frac{ax^3-ax+b}{1-x^2}\;dx$ . Input constant of integration as $C$ .
Luis's copy of Indefinite integral by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$ . Solution involves the inverse trigonometric function $\arcsin$ .
Luis's copy of Indefinite integral by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$ . Solution involves the inverse trigonometric function $\arcsin$ .
Luis's copy of Indefinite integral by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$
Luis's copy of Implicit differentiation
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Question in MAT333 by Luis Hernandez and 1 other

Implicit differentiation.

Given $x^2+y^2+ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$ .
Luis's copy of Integration by parts
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Question in MAT333 by Luis Hernandez and 1 other

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$ , find $g(x)$ . Find $h(x)$ , $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$ .
Luis's copy of Integration by parts
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx$ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$
Luis's copy of Integration by parts
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Question in MAT333 by Luis Hernandez and 1 other

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 $g(x)$ .
Luis's copy of Integration by partial fractions
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d$ .
Luis's copy of Integration by partial fractions
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d$ .
Luis's copy of Integration by partial fractions
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Question in MAT333 by Luis Hernandez and 1 other

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.
Luis's copy of Integration by partial fractions
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Question in MAT333 by Luis Hernandez and 1 other

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.
Luis's copy of Integrate algebraic fraction
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$ . Enter the constant of integration as $C$ .
Luis's copy of Integration: Indefinite integral by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$
Luis's copy of Integration: Indefinite integral by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$
Luis's copy of Integration: Indefinite integral by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$
Luis's copy of Integration: Indefinite integral by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$
Luis's copy of Integration: Indefinite integral
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$
Luis's copy of Integration: Indefinite integral
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$ .
Luis's copy of Integration: Indefinite integral
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle \int ax ^ m+ bx^{c/n}\;dx$ .
Luis's copy of Integration of fraction with power in denominator
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Question in MAT333 by Luis Hernandez and 1 other

$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$ for a polynomial $g(x)$ . Find $g(x)$ .
Luis's copy of Integration by substitution
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Question in MAT333 by Luis Hernandez and 1 other

Find $\displaystyle I=\int \frac{2 a x + b} {a x ^ 2 + b x + c}\;dx$ by substitution or otherwise.
Luis's copy of Limiting speed given acceleration function,
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Question in MAT333 by Luis Hernandez and 1 other

An object moves in a straight line, acceleration given by:

$\displaystyle f(t)=\frac{a}{(1+bt)^n}$ . The object starts from rest. Find its maximum speed.
Luis's copy of Quotient rule - differentiate quadratic over quadratic
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Question in MAT333 by Luis Hernandez and 1 other

The derivative of $\displaystyle \frac{ax^2+b}{cx^2+d}$ is $\displaystyle \frac{g(x)}{(cx^2+d)^2}$ . Find $g(x)$ .
Luis's copy of Quotient rule - differentiate linear over square root
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Question in MAT333 by Luis Hernandez and 1 other

The derivative of $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$ . Find $g(x)$ .
Luis's copy of Quotient rule - differentiate linear over quadratic
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Question in MAT333 by Luis Hernandez and 1 other

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$ . Find $g(x)$ .
Luis's copy of Quotient rule - differentiate linear over quadratic
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Question in MAT333 by Luis Hernandez and 1 other

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$ . Find $g(x)$ .
Luis's copy of Quotient rule - differentiate exponential over exponential
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Question in MAT333 by Luis Hernandez and 1 other

The derivative of $\displaystyle \frac{a+be^{cx}}{b+ae^{cx}}$ is $\displaystyle \frac{pe^{cx}} {(b+ae^{cx})^2}$ . Find $p$ .

Luis Hernandez

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