Reminders Coverage Sample questions

Mathematics 54First Long Examination

02 September 2014, Tuesday

Discussion class room and time

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Reminders Coverage Sample questions

Math 53 methods of integration!!!

Integration by parts:∫

udv = uv−∫

v du

Trigonometric integralsfactoring out sinu, cosu, sec2 u, csc2 u, secu tanu, cscucotuidentities

Trigonometric substitutionp

a2 −u2p

a2 +u2p

u2 −a2

Partial fractionsdistinct linear factors, repeated linear factorsdistinct quadratic factors, repeated quadratic factors

Improper integralsinfinite limit/s of integrationinfinite discontinuity at point on the interval of integration

Differential equationsgeneral/particular solution of a separable differential equationapplications (e.g. orthogonal trajectories, exponentialgrowth/decay) 2 / 9

Reminders Coverage Sample questions

Find the antiderivative:∫

(cosx+csc2 x)2 dx

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Reminders Coverage Sample questions

Evaluate the integral∫ p

π

0x3 sinx2 dx.

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Reminders Coverage Sample questions

Find the antiderivative:∫

xpx2 −4x

dx

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Reminders Coverage Sample questions

Find the antiderivative:∫

x3 +3x−2

x2(x2 +3)dx

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Reminders Coverage Sample questions

Determine whether the improper integral is convergent or

divergent:∫ +∞

−∞e−|x| dx

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Reminders Coverage Sample questions

Find the equation of the curve for whichd2y

dx2 = 1

(2x+4)2 and that is

tangent to the x-axis at the point(− 3

2,0

).

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Reminders Coverage Sample questions

Find the orthogonal trajectories to the family of curves

y = 1px−k

, k < x.

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