IB Mathematics Analysis and Approaches Topic AHL…

Express $\frac{4x^2+5x+3}{(x+1)(x^2+2x+2)}$ in partial fractions.

Hence, evaluate $\int_0^1 \frac{4x^2+5x+3}{(x+1)(x^2+2x+2)} \, dx$ .

Let $y=\arcsin \left(\frac{x}{3}\right)$ for .

Prove that $(\sec x+\tan x)^{2}=\frac{1+\sin x}{1-\sin x}$ .

The identity $\frac{4x^2+4x+6}{(x-2)(x+1)(x+3)}\equiv\frac{A}{x-2}+\frac{B}{x+1}+\frac{C}{x+3}$ holds, where .

Let $g(y) = \frac{1}{(y-2)(y+1)}$ for .

Find $\frac{d y}{d x}$ .

Find $\int_{-3}^{3} \arcsin \left(\frac{x}{3}\right) d x$ .

Hence, find $\int_{0}^{\frac{\pi}{3}} 3(\sec x+\tan x)^{2}\,dx$ , expressing the answer in the form $a+b\sqrt{3}+c\pi$ , where $a,b,c\in\mathbb{Z}$ .

Find the values of $A$ , $B$ , and $C$ .

Hence, express $\frac{4x^2+4x+6}{(x-2)(x+1)(x+3)}$ as the sum of partial fractions.

Evaluate the integral $\int \frac{4x^2+4x+6}{(x-2)(x+1)(x+3)}\,dx$ .

Find $\int g(y) \, \mathrm{d}y$ .

IB Mathematics Analysis and Approaches AHL 5.15—further Derivatives and Indefinite Integration of These, Partial Fractions Questionbank