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

What these 13 flashcards ask

\frac{d}{dx}\arcsin x?
\frac{d}{dx}\arccos x?
\frac{d}{dx}\arctan x?
\frac{d}{dx}\arcsin(3x)?
\frac{d}{dx}\left[\frac12\arctan(x^2)\right]?
Derivative of \arcsin x+x\sqrt{1-x^2}?
\int\frac{1}{\sqrt{a^2-x^2}}\,dx?
\int\frac{1}{a^2+x^2}\,dx?
\int\frac{1}{\sqrt{9-4x^2}}\,dx?
Substitution for \sqrt{a^2-x^2}?
Substitution for a^2+x^2?
What must you do to the limits after a substitution?
Use of \cos^2\theta in integration?

Exam questions on Inverse trigonometric functions: differentiation and integration

Let $f(x)=\frac12\arctan\left(x^2\right)$ .
Hence find the exact value of $\int_0^1\frac{x}{1+x^4}\,dx$ .2 marks
Let $y=\arcsin x+x\sqrt{1-x^2}$ for $-1<x<1$ .
Hence find the exact value of $\int_0^{1/2}\sqrt{1-x^2}\,dx$ .2 marks
Let $f(x)=\frac{1}{\sqrt{9-4x^2}}$ and $g(x)=\frac{1}{9+4x^2}$ .
Use the substitution $x=\frac32\sin\theta$ to show that $\int_0^{3/4}f(x)\,dx=\frac\pi{12}$ .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).