Inverse trigonometric functions: differentiation and integrationAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

\dfrac{d}{dx}\arcsin x?
\dfrac{d}{dx}\arccos x?
\dfrac{d}{dx}\arctan x?
\dfrac{d}{dx}\arctan u for a function u(x)?
\displaystyle\int\frac{1}{\sqrt{a^2-x^2}}\,dx?
\displaystyle\int\frac{1}{a^2+x^2}\,dx?
Substitution for \sqrt{a^2-x^2}?
Substitution for a^2+x^2?
Why is \cos y=\sqrt{1-x^2} (positive root) when y=\arcsin x?
Where is the derivative of \arcsin x undefined?
How do you integrate \dfrac{1}{x^2-4x+13}?
How do you integrate \dfrac{x+1}{x^2-4x+13}?

Exam questions on Inverse trigonometric functions: differentiation and integration

Let $\mathrm{f}(x)=\arctan(3x)$ .
Find the equation of the tangent to the curve $y=\mathrm{f}(x)$ at the point where $x=\dfrac13$ .2 marks
The curve $C$ has equation $y=\arccos\left(\dfrac{x}{2}\right)$ for $-2\le x\le2$ .
Find the values of $x$ for which the gradient of $C$ is undefined, and describe the tangent to $C$ at these points.2 marks
Let $I=\int_0^1\dfrac{1}{\sqrt{4-x^2}}\,dx$ and $J=\int_0^1\sqrt{4-x^2}\,dx$ .
Find the exact value of $I$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).