Mathematical Methods — Question 11

QCAA 2025, Paper 1 · 5 marks

Q11 · 2025 · Technology-freeSimple familiar5 marks

QUESTION 11 (5 marks)

Determine the following integrals.

\displaystyle\int\frac1x\,dx

[1 mark]

\displaystyle\int e^{4x}\,dx

[1 mark]

\displaystyle\int x(x^3+8)\,dx

[3 marks]

WORKED SOLUTION

5 marks

ANSWER

\ln(x)+c

; b)

\frac14e^{4x}+c

; c)

\frac{x^5}{5}+4x^2+c

Worked solution

\int\frac1x\,dx=\ln(x)+c.

\int e^{4x}\,dx=\frac14e^{4x}+c.

\int x(x^3+8)\,dx=\int(x^4+8x)\,dx=\frac{x^5}{5}+4x^2+c.

correctly applies the integration rule

[1 mark]

correctly integrates the exponential term

[1 mark]

correctly expands the brackets

[1 mark]

integrates the power of 4 term

[1 mark]

integrates the linear term

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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