How many SCSA Mathematics Specialist questions cover Integration and applications of integration?

AusGrader has 67 SCSA Mathematics Specialist questions on Integration and applications of integration, all with instant AI grading and detailed marking feedback.

SCSA (WACE) Mathematics Specialist — Integration and applications of integration Questions & Answers | AusGrader

Q3

2022

QCAA

Paper 2

1 mark

Q3

1 mark

Determine the solution of the differential equation $\frac{dy}{dx} = \frac{\sin(2x)}{\cos(2x)}$ given $y=0$ when $x=\frac{\pi}{5}$ .

A

$y = -2\ln|\cos(2x)| - 2.35$

B

$y = -2\ln|\cos(2x)| + 2.35$

C

$y = -\frac{1}{2}\ln|\cos(2x)| - 0.59$

D

$y = -\frac{1}{2}\ln|\cos(2x)| + 0.59$

Q2

2024

QCAA

Paper 1

1 mark

Q2

1 mark

Given that $\frac{A}{x-2} + \frac{3}{x} = \frac{x-6}{x(x-2)}$ , determine the value of $A$ .

A

$-4$

B

$-2$

C

$2$

D

$4$

Q1

2020

SCSA

Paper 1

3 marks

Q1

3 marks

Evaluate exactly $\int_0^\pi (4\cos^2 x - \sin x) \, dx$ .

\begin{align*} \int_0^\pi (4\cos^2 x - \sin x) dx &= \int_0^\pi (2(1+\cos 2x) - \sin x) dx\\ &= \int_0^\pi (2 + 2\cos 2x - \sin x) dx \quad ... (1)\\ &= \left[ 2x + \sin 2x + \cos x \right]_0^\pi\\ &= (2\pi + 0 - 1) - (0 + 0 + 1)\\ &= 2\pi - 2 \end{align*}

Marking Criteria

Descriptor	Marks
uses the cosine double angle identity to obtain integrand (1)	1
anti-differentiates the integrand correctly	1
evaluates the definite integral correctly	1

Q4

2022

SCSA

Paper 1

8 marks

Q4a

3 marks

Function $f(x) = \frac{5(x + 1)}{(x - 1)(x^2 + 3x + 1)}$ can be expressed in the form $\frac{a}{x - 1} + \frac{bx + c}{x^2 + 3x + 1}$ .

Determine the value of the constants $a$ , $b$ and $c$ .

\begin{aligned} \frac{5x+5}{(x-1)(x^2+3x+1)} &= \frac{a(x^2+3x+1) + (x-1)(bx+c)}{(x-1)(x^2+3x+1)} \\ &= \frac{(a+b)x^2 + (3a-b+c)x + (a-c)}{(x-1)(x^2+3x+1)} \end{aligned}

Equating coefficients:

\begin{align*} a+b &= 0\\ 3a-b+c &= 5\\ a-c &= 5 \end{align*}

Solving gives $a=2$ , $b=-2$ , $c=-3$

\begin{align*} \text{i.e. }\frac{5(x+1)}{(x-1)(x^2+3x+1)} = \frac{2}{x-1} - \frac{(2x+3)}{x^2+3x+1} \end{align*}

Marking Criteria

Descriptor	Marks
forms the correct expression for the equivalent numerator	1
equates coefficients correctly to form 3 linear equations	1
solves correctly to determine $a$ , $b$ and $c$	1

Q4b

5 marks

Hence determine $\int \frac{10x + 10}{(x - 1)(x^2 + 3x + 1)} \, dx$ .

\begin{aligned} \int \frac{10x+10}{(x-1)(x^2+3x+1)} dx &= 2 \int \frac{5x+5}{(x-1)(x^2+3x+1)} dx \\ &= \int \frac{4}{x-1} - \frac{2(2x+3)}{x^2+3x+1} dx \\ &= 4\ln|x-1| - 2\ln|x^2+3x+1| + k \\ &= \ln\left( \frac{(x-1)^4}{(x^2+3x+1)^2} \right) + k \end{aligned}

Marking Criteria

Descriptor	Marks
expresses the given integrand as double $f(x)$	1
writes the integrand correctly in terms of the partial fractions	1
anti-differentiates $\frac{a}{x-1}$ correctly using the absolute value of a natural logarithm	1
anti-differentiates $\frac{bx+c}{x^2+3x+1}$ correctly	1
uses a constant of integration	1

Q3

2022

SCSA

Paper 1

5 marks

Q3

5 marks

By using one or more of the following identities:

$\cos^2 x + \sin^2 x = 1$
$\cos 2x = \cos^2 x - \sin^2 x$
$\sin 2x = 2 \sin x \cos x$

evaluate exactly $\int_0^{\frac{\pi}{2}} (\sin x + \cos x)^2 \, dx$ .

\begin{aligned} \int_0^{\frac{\pi}{2}} (\sin x + \cos x)^2 dx &= \int_0^{\frac{\pi}{2}} (\sin^2 x + 2\sin x \cos x + \cos^2 x) dx \\ &= \int_0^{\frac{\pi}{2}} ((\sin^2 x + \cos^2 x) + 2\sin x \cos x) dx \\ &= \int_0^{\frac{\pi}{2}} (1 + \sin 2x) dx \\ &= \left[ x - \frac{\cos 2x}{2} \right]_0^{\frac{\pi}{2}} \\ &= \left( \frac{\pi}{2} - \frac{\cos \pi}{2} \right) - \left( 0 - \frac{\cos 0}{2} \right) \\ &= \frac{\pi}{2} + \frac{1}{2} - \left( -\frac{1}{2} \right) \\ &= \frac{\pi}{2} + 1 \end{aligned}

Marking Criteria

Descriptor	Marks
expands the integrand correctly	1
uses the Pythagorean identity $\sin^2 x + \cos^2 x = 1$	1
uses double angle identity for $\sin 2x$	1
anti-differentiates the trigonometric function correctly	1
evaluates correctly using exact trigonometric values	1

SCSA Mathematics Specialist Integration and applications of integration

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