Harder Substitution (Edexcel A-Level Mathematics): Flashcards

It's $\frac{x^{n+1}}{n+1} + C$, for $n \neq -1$.

The derivative is $du/dx = 2x$.

$dx = \frac{du}{2x}$ after differentiating.

It's $\frac{1}{3}(1 + x^2)^{3/2} + C$, via substitution.

$\int u^{1/2} du = \frac{2}{3} u^{3/2} + C$.

$\frac{1}{3} (1 + x^2)^{3/2} + C$

Let $u = 1 + \cos(2x)$ for substitution.

Use $x = \sec(\theta)$ for substitution.

$dx = \sec(\theta)\tan(\theta) d\theta$.

It simplifies to $\frac{(\ln(x))^2}{2} + C$.