Pure B: Advanced Calculus

Topic overview

Integration by substitution, integration by parts, infinite series, differential equations

Key concepts

Verify STEP 2/3 separately. Complex numbers and 2×2 matrices may belong to STEP 2; De Moivre, general complex roots, planes and vector products, etc. belong to STEP 3.

Key points

◆Follow the prerequisite knowledge requirements given by the examination body; new concepts may only be used when the question provides sufficient guidance.
◆The examples on this site are training with a limited scope and do not represent full syllabus coverage or official difficulty.

Worked examples

Try each example yourself before checking the explanation.

1Written examples within the scope

Show, by integrating by parts twice, that $\displaystyle\int e^x \cos x\,dx = \frac{e^x(\sin x+\cos x)}{2} + C$ .

Evaluate $\displaystyle\int_0^{\pi} e^x \cos x\,dx$ .

By differentiating the result in (i) or otherwise, find $\displaystyle\int e^x \sin x\,dx$ .

Find the total area enclosed between $y = e^x \cos x$ and the $x$ -axis for $x \in [0,\, 2\pi]$ .

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