CMA01

Started: 30 Sep at 18:50

Quiz instructions

This Computer-Marked Assignment (CMA) covers topics from the first 3 lectures of this MTH108 course, with a heavy emphasis on techniques of integration.

There are 20 MCQ questions, each worth 5 marks.

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Let $LaTeX: f\left(x\right)=\ln\left(1+x^2\right)$ $f$ ( x ) = ln ⁡ ( 1 + x 2 ) for $LaTeX: x\le0$ $x$ ≤ 0 .

Determine the rule for $LaTeX: f^{-1}\left(x\right)$ $f$ − 1 ( x ) .

Group of answer choices

$LaTeX: f^{-1}\left(x\right)=-\sqrt{e^x-1}$ $f^{- 1} (x) = - \sqrt{e^{x} - 1}$

$LaTeX: f^{-1}\left(x\right)=\frac{2x}{1+x^2}$ $f^{- 1} (x) = \frac{2 x}{1 + x^{2}}$

$LaTeX: f^{-1}\left(x\right)=\frac{1}{\ln\left(1+x^2\right)}$ $f^{- 1} (x) = \frac{1}{\ln (1 + x^{2})}$

$LaTeX: f^{-1}\left(x\right)=\sqrt{e^x-1}$ $f^{- 1} (x) = \sqrt{e^{x} - 1}$

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Let $LaTeX: F\left(x\right)=\arcsin\sqrt{x}$ $F$ ( x ) = arcsin ⁡ x for LaTeX: 0<x<1 $0$ < x < 1. Compute $LaTeX: F'\left(x\right)$ $F$ ′ ( x ) for $0$ < x < 1.

Group of answer choices

$LaTeX: \frac{1}{\sqrt{x-x^2}}$ $\frac{1}{\sqrt{x - x^{2}}}$

$LaTeX: \frac{1}{2\sqrt{x-x^2}}$

\frac{1}{2 \sqrt{x - x^{2}}}

\frac{1}{2 \sqrt{x - x^{2}}}

\frac{1}{2 \sqrt{x - x^{2}}}

$LaTeX: \frac{\arccos\sqrt{x}}{2\sqrt{x}}$ $\frac{\arccos \sqrt{x}}{2 \sqrt{x}}$

$LaTeX: \frac{1}{\sqrt{1-x}}$

\frac{1}{\sqrt{1 - x}}

\frac{1}{\sqrt{1 - x}}

\frac{1}{\sqrt{1 - x}}

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Compute the indefinite integral

$LaTeX: \int\left(-\frac{6}{x^2}-\frac{5}{\cos^2x}\right)dx$ $∫$ ( − 6 x 2 − 5 cos 2 ⁡ x ) d x

Group of answer choices

$LaTeX: \frac{6}{x}+\frac{5}{\cos x}+C$

\frac{6}{x} + \frac{5}{\cos x} + C

\frac{6}{x} + \frac{5}{\cos x} + C

$LaTeX: \frac{6}{x}-5\tan x+C$

\frac{6}{x} - 5 \tan x + C

\frac{6}{x} - 5 \tan x + C

$LaTeX: \frac{6}{x}+5\tan x+C$

\frac{6}{x} + 5 \tan x + C

\frac{6}{x} + 5 \tan x + C

$LaTeX: -\frac{6}{x}-5\tan x+C$

- \frac{6}{x} - 5 \tan x + C

- \frac{6}{x} - 5 \tan x + C

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Compute the indefinite integral $LaTeX: \int\left(\frac{x-1}{x^2+1}\right)dx$ $∫$ ( x − 1 x 2 + 1 ) d x

Group of answer choices

$LaTeX: \frac{\ln\left(1+x^2\right)}{2}-\arctan x+C$

\frac{\ln (1 + x^{2})}{2} - \arctan x + C

\frac{\ln (1 + x^{2})}{2} - \arctan x + C

\frac{\ln (1 + x^{2})}{2} - \arctan x + C

\frac{\ln (1 + x^{2})}{2} - \arctan x + C

$LaTeX: \ln\left(1+x^2\right)-\arctan x+C$ $\ln (1 + x^{2}) - \arctan x + C$

$LaTeX: \arctan\:\left(1+x^2\right)$ $\arctan (1 + x^{2})$ $LaTeX: +\:C$ $+ C$

$LaTeX: \frac{\arctan\left(1+x^2\right)}{2}+C$ $\frac{\arctan (1 + x^{2})}{2} + C$

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Compute the indefinite integral $LaTeX: \int\sin^3xdx$

∫

sin 3 ⁡ x d x

Group of answer choices

$LaTeX: \frac{\cos3x}{3}-\cos x+C$

\frac{\cos 3 x}{3} - \cos x + C

\frac{\cos 3 x}{3} - \cos x + C

$LaTeX: \frac{\sin4x}{4}-\sin4x+C$

\frac{\sin 4 x}{4} - \sin 4 x + C

\frac{\sin 4 x}{4} - \sin 4 x + C

$LaTeX: \frac{\cos^3x}{3}-\cos x+C$

\frac{\cos^{3} x}{3} - \cos x + C

\frac{\cos^{3} x}{3} - \cos x + C

$LaTeX: \frac{\sin^4x}{4}+C$

\frac{\sin^{4} x}{4} + C

\frac{\sin^{4} x}{4} + C

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In the computation of the indefinite integral

$LaTeX: \int\left(\ln x\right)^2\:dx$ $∫$ ( ln ⁡ x ) 2 d x

which of the following techniques is used?

Group of answer choices

Integration by Partial Fractions

Product Rule

Integration by Substitution

Integration by Parts

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In the computation of the indefinite integral

$LaTeX: \int\frac{2x^3-3x^2+4x}{x^2-2x+2}dx$ $∫$ 2 x 3 − 3 x 2 + 4 x x 2 − 2 x + 2 d x

which of the following techniques is used?

Group of answer choices

Quotient Rule

Long Division of Polynomials

Integration by Parts

Integration by Partial Fractions

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In the solution of the integral

$LaTeX: \int\frac{x}{\sqrt{1-x^4}}\:dx$ $∫$ x 1 − x 4 d x

which of the following substitutions is used?

Group of answer choices

LaTeX: u=x $u = x$

LaTeX: u=1-x^4 $u = 1 - x^{4}$

LaTeX: u=x^4 $u = x^{4}$

LaTeX: u=x^2 $u = x^{2}$

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In the computation of the indefinite integral

$LaTeX: \int\sin^2\left(3x\right)dx$ $∫$ sin 2 ⁡ ( 3 x ) d x

which of the following trigonometric identities is used?

Group of answer choices

$LaTeX: \sin^2\theta+\cos^2\theta\:=1$ $\sin^{2} θ + \cos^{2} θ = 1$ $\sin^{2} θ + \cos^{2} θ = 1$

$LaTeX: \cos2\theta=\cos^2\theta-\sin^2\theta$ $\cos 2 θ = \cos^{2} θ - \sin^{2} θ$ $\cos 2 θ = \cos^{2} θ - \sin^{2} θ$

$LaTeX: \cos2\theta=1-2\:\sin^2\theta$ $\cos 2 θ = 1 - 2 \sin^{2} θ$ $\cos 2 θ = 1 - 2 \sin^{2} θ$

$LaTeX: \cos2\theta=1-\sin^2\theta$ $\cos 2 θ = 1 - \sin^{2} θ$ $\cos 2 θ = 1 - \sin^{2} θ$

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If $LaTeX: \frac{8x+5}{3x^{2}-5x-2}=\frac{A}{3x+1}+\frac{B}{x-2}$

8

x + 5 3 x 2 − 5 x − 2 = A 3 x + 1 + B x − 2, then

Group of answer choices

$LaTeX: \text{A=1, B=3}$

A=1, B=3

A=1, B=3

$LaTeX: A=-1,\:B=3$

A = - 1, B = 3

A = - 1, B = 3

$LaTeX: \text{A=2, B=3}$

A=2, B=3

A=2, B=3

$LaTeX: \text{A=2, B=0}$

A=2, B=0

A=2, B=0

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Evaluate the definite integral $LaTeX: \int^{\pi}_0\left(1+\tan x\:\cos x\right)dx$ $∫$ 0 π ( 1 + tan ⁡ x cos ⁡ x ) d x

Group of answer choices

$LaTeX: -\pi$

- π

- π

- π

- π

$LaTeX: \pi$

π

π

π

π

$LaTeX: \pi-2$ $π - 2$

$LaTeX: \pi+2$

π + 2

π + 2

π + 2

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Evaluate the definite integral $LaTeX: \int_0^6\frac{1}{x^2+36}dx$ $∫$ 0 6 1 x 2 + 36 d x.

Group of answer choices

1

1

1

$LaTeX: \frac{\pi}{2}$

\frac{π}{2}

\frac{π}{2}

\frac{π}{2}

$LaTeX: \frac{\pi}{6}$ $\frac{π}{6}$

$LaTeX: \frac{\pi}{24}$

\frac{π}{24}

\frac{π}{24}

\frac{π}{24}

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Evaluate the definite integral $LaTeX: \int^1_0xe^{-2x}\:dx$ $∫$ 0 1 x e − 2 x d x

Group of answer choices

$LaTeX: \frac{1-e^{-2}}{3}$

\frac{1 - e^{- 2}}{3}

\frac{1 - e^{- 2}}{3}

\frac{1 - e^{- 2}}{3}

\frac{1 - e^{- 2}}{3}

$LaTeX: \frac{1-3e^{-2}}{4}$

\frac{1 - 3 e^{- 2}}{4}

\frac{1 - 3 e^{- 2}}{4}

\frac{1 - 3 e^{- 2}}{4}

\frac{1 - 3 e^{- 2}}{4}

$LaTeX: \frac{1-3e^{-2}}{3}$ $\frac{1 - 3 e^{- 2}}{3}$

$LaTeX: \frac{1-e^{-2}}{4}$ $\frac{1 - e^{- 2}}{4}$

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Compute the definite integral $LaTeX: \int^1_0\cos\pi x\:dx$

∫

0 1 cos ⁡ π x d x

Group of answer choices

$LaTeX: \frac{1}{\pi}$

\frac{1}{π}

\frac{1}{π}

\frac{1}{π}

$LaTeX: \frac{2}{\pi}$

\frac{2}{π}

\frac{2}{π}

\frac{2}{π}

$LaTeX: -\frac{1}{\pi}$

- \frac{1}{π}

- \frac{1}{π}

- \frac{1}{π}

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Compute the definite integral $LaTeX: \int^1_0\left|\cos\pi x\right|dx$

∫

0 1 | cos ⁡ π x | d x.

Group of answer choices

$LaTeX: \pi$

π

π

π

π

$LaTeX: \frac{2}{\pi}$

\frac{2}{π}

\frac{2}{π}

\frac{2}{π}

\frac{2}{π}

$LaTeX: \frac{1}{\pi}$

\frac{1}{π}

\frac{1}{π}

\frac{1}{π}

\frac{1}{π}

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Evaluate the definite integral $LaTeX: \int^{\frac{\pi}{3}}_0\tan^3x\:dx$ $∫$ 0 π 3 tan 3 ⁡ x d x.

Group of answer choices

$LaTeX: \frac{\pi}{6}$ $\frac{π}{6}$ $\frac{π}{6}$

$LaTeX: \frac{3}{2}-\ln2$

\frac{3}{2} - \ln 2

\frac{3}{2} - \ln 2

\frac{3}{2} - \ln 2

\frac{3}{2} - \ln 2

\frac{3}{2} - \ln 2

\frac{3}{2} - \ln 2

$LaTeX: 1-\ln2$

1 - \ln 2

1 - \ln 2

1 - \ln 2

1 - \ln 2

1 - \ln 2

1 - \ln 2

$LaTeX: \frac{\ln2}{2}-\frac{3}{4}$ $\frac{\ln 2}{2} - \frac{3}{4}$

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Evaluate the definite integral $LaTeX: \int^1_0\arctan x\:dx$ $∫$ 0 1 arctan ⁡ x d x.

Group of answer choices

$LaTeX: \frac{\pi}{4}-\frac{\ln2}{2}$

\frac{π}{4} - \frac{\ln 2}{2}

\frac{π}{4} - \frac{\ln 2}{2}

\frac{π}{4} - \frac{\ln 2}{2}

\frac{π}{4} - \frac{\ln 2}{2}

\frac{π}{4} - \frac{\ln 2}{2}

$LaTeX: \frac{\pi}{4}+\frac{\ln2}{2}$

\frac{π}{4} + \frac{\ln 2}{2}

\frac{π}{4} + \frac{\ln 2}{2}

\frac{π}{4} + \frac{\ln 2}{2}

\frac{π}{4} + \frac{\ln 2}{2}

\frac{π}{4} + \frac{\ln 2}{2}

$LaTeX: \frac{\ln2}{2}-\frac{1}{4}$ $\frac{\ln 2}{2} - \frac{1}{4}$

$LaTeX: \frac{\pi}{4}-\frac{1}{2}$ $\frac{π}{4} - \frac{1}{2}$

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Let f be a continuous function on [-1, 1]. Define F on [-1, 1] by

$LaTeX: F\left(x\right)=\int_0^xf\left(t\right)dt$ $F$ ( x ) = ∫ 0 x f ( t ) d t.

Compute F(0)

Group of answer choices

Does not exist

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Let f be a continuous function on [-1, 1]. Define F on [-1, 1] by

$LaTeX: F\left(x\right)=\int_0^xf\left(t\right)dt$ $F$ ( x ) = ∫ 0 x f ( t ) d t. If $LaTeX: F'\left(x\right)=e^x+2x+1$ $F$ ′ ( x ) = e x + 2 x + 1 for x in (-1, 1)

Evaluate f(0).

Group of answer choices

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Let f be a function on $LaTeX: \left(-\infty,\:\infty\right)$ $($ − ∞ , ∞ ) such that $LaTeX: \int f\left(x\right)dx=x^2e^{2x}+C$ $∫$ f ( x ) d x = x 2 e 2 x + C. Compute f(x).