• 1. Watch the video:

Key Points

• 1. Watch the video:

• 2. Watch this special case of definite integrals.

Key Points

• 1. You should be able to use the results given in your formula booklet to differentiate and integrate various functions. Standard integrals can be combined with a linear substitution: if \( \int f(x)\,dx = F(x) \), then \[ \int f(ax + b)\,dx = \frac{1}{a}F(ax + b). \] In particular, \[ \int \frac{1}{\sqrt{a^2 - (x + b)^2}}\,dx = \arcsin\!\left(\frac{x + b}{a}\right) + c, \] \[ \int \frac{1}{a^2 + (x + b)^2}\,dx = \frac{1}{a}\arctan\!\left(\frac{x + b}{a}\right) + c. \]

• 2. You should be able to use integration by substitution: differentiate the substitution to express \( dx \) in terms of \( du \); replace all occurrences of \( x \) by the relevant expression in terms of \( u \); change the limits from \( x \) to \( u \); and simplify as much as possible before integrating.

• 3. You should be able to use integration by parts: \[ \int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx \]