CAIE A-Level Mathematics 3.5.5 Inverse tangent

Practise integration by parts for products involving x, x^2, exponentials, ln x and tan^-1 expressions, including exact areas.

Exam points

reduce the remaining integral after parts to a standard ln, tan^-1 or exponential form
use integration by parts inside area calculations with given bounds and exact answers

[Maximum number: 5]

Show that $\int_{0}^{\sqrt{3}} x \tan ^{-1} x \mathrm{~d} x=\frac{2}{3} \pi-\frac{1}{2} \sqrt{3}$ .

Integrate by parts and reach $a x^{2} \tan ^{-1} x+b \int \frac{x^{2}}{1+x^{2}} \mathrm{~d} x$

M1

Obtain $\frac{1}{2} x^{2} \tan ^{-1} x-\frac{1}{2} \int \frac{x^{2}}{1+x^{2}} \mathrm{~d} x$ , or equivalent

A1

Obtain complete indefinite integral $\frac{1}{2}\left(x^{2} \tan ^{-1} x-x+\tan ^{-1} x\right)$ , or equivalent

A1

Substitute limits having integrated twice

M1

Obtain the given answer correctly

A1

AG

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3.5.5—Inverse tangent question 1