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Julie's copy of Second order differential equations 2
Draft
Question in Julie's workspace by Julie Crowley

Method of undermined coefficients:

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$ . (Equal roots example). Includes an interactive plot.

rebelmaths
Julie's copy of Second order differential equations 1
Draft
Question in Julie's workspace by Julie Crowley

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+(a^2+b^2)y=0,\;y(0)=1$ and $y'(0)=c$ .

rebelmaths
Indefinite integral by substitution
Draft
Question in Bill's workspace by Bill Foster

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$ . Solution involves the inverse trigonometric function $\arcsin$ .
Indefinite integral by substitution
Draft
Question in Bill's workspace by Bill Foster

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$
Indefinite integral by substitution
Draft
Question in Bill's workspace by Bill Foster

Find $\displaystyle \int\frac{ax+b}{(1-x^2)^{1/2}} \;dx$ . Solution involves inverse trigonometric functions.
Leicester: Integration 3
Draft
Question in Bill's workspace by Bill Foster

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

Also two other questions on integrating by parts.
Indefinite integral
Draft
Question in Bill's workspace by Bill Foster

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $\sin(ax+b) e ^ {nx}$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $(a+bx) ^ {m} \sin(nx)$
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $x^m\cos(ax+b)$
Indefinite integral by substitution
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$
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 $C$ .
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 $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$ . 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 $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$ .
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 $\arctan$ .
Integrate algebraic 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 $C$ .
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 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.
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 $\arctan$ .
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 parts
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx$ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$
Differentiate product of trig function and binomial
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Differentiate $(a+bx) ^ {m} \sin(nx)$
Differentiation: Product rule
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Differentiate $x^m\cos(ax+b)$
Differentiation: Product rule
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Differentiate $\sin(ax+b) e ^ {nx}$ .
Second order ODE with constant coefficients and boundary conditions
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+(a^2+b^2)y=0,\;y(0)=1$ and $y'(0)=c$ .
Solve a second order ODE with repeated roots,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$ . (Equal roots example).
Julie's copy of Indefinite integral by substitution
Draft
Question in Julie's workspace by Julie Crowley

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$
Julie's copy of Indefinite integral
Draft
Question in Julie's workspace by Julie Crowley

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$ .
Indefinite integral by substitution
Draft
Question in Julie's workspace by Julie Crowley

Find $\displaystyle \int\frac{ax+b}{(1-x^2)^{1/2}} \;dx$ . Solution involves inverse trigonometric functions.

rebelmaths

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