Integration Past Paper Questions — 9709 Paper 3

Q112025 Feb/Mar·P326MMediumIntegration

Find the exact value of $\int_0^\pi x^2 \cos \frac{1}{3}x\,dx$ .

Q32025 May/Jun·P334MMedium-EasyIntegration

Find the exact value of

\int_{\frac{1}{5}\pi}^{\frac{1}{4}\pi} 3\cos^2 5x\,dx

Q12025 Oct/Nov·P314MMedium-EasyIntegration

Find the exact value of $\int_1^2 \ln 3x \, dx$ . Give your answer in the form $a + \ln b$ , where $a$ and $b$ are integers.

Q82025 Oct/Nov·P318MMedium-HardDifferential Equations

The variables $x$ and $y$ satisfy the differential equation

(x^2 + 1)\frac{dy}{dx} = kxe^{2y}

where $k$ is a constant. It is given that $y = 0$ when $x = 0$ and that $y = -\frac{1}{2}$ when $x = 1$ .

Solve the differential equation and find the exact value of $y$ when $x = \sqrt{3}$ .

Q42025 Oct/Nov·P326MMediumIntegration

Find the exact value of $\int_0^1 x \tan^{-1} x \, dx$ .

Q62025 Oct/Nov·P336MMediumIntegration

Find the exact value of $\int_0^{\frac{1}{6}\pi} x^2 \sin 2x \, dx$ .

Q72025 Oct/Nov·P352 partsMedium-EasyIntegration

(a)

Use the substitution $u = x^2 - 3$ to show that

\int_{\sqrt{7}}^{\sqrt{12}} \frac{4x^3}{\sqrt{x^2 - 3}} \,dx = \int_a^b \frac{2(u + 3)}{\sqrt{u}} \,du

where $a$ and $b$ are values to be found.

(b)

Hence, find the exact value of $\int_{\sqrt{7}}^{\sqrt{12}} \frac{4x^3}{\sqrt{x^2 - 3}} \,dx$ .

Q112024 Feb/Mar·P329MMedium-HardDifferential Equations

The variables $y$ and $\theta$ satisfy the differential equation

(1 + y)(1 + \cos 2\theta)\frac{dy}{d\theta} = e^{3y}.

It is given that $y = 0$ when $\theta = \frac{1}{4}\pi$ .

Solve the differential equation and find the exact value of $\tan\theta$ when $y = 1$ .

Q82024 May/Jun·P317MMedium-HardIntegration

Use the substitution $u = 1 - \sin x$ to find the exact value of

\int_{\pi}^{\frac{3}{2}\pi} \frac{\sin 2x}{\sqrt{1 - \sin x}} \,dx

Give your answer in the form $a + b\sqrt{2}$ where $a$ and $b$ are rational numbers to be determined.

Q112024 May/Jun·P339MMedium-HardIntegration

Use the substitution $2x = \tan\theta$ to find the exact value of

\int_0^{\frac{1}{2}} \frac{12}{(1 + 4x^2)^2}\,dx

Give your answer in the form $a + b\pi$ , where $a$ and $b$ are rational numbers.