CAIE A-Level Mathematics A2 3.5.5 Inverse Tangent Questions

Practise integration by parts, including inverse trigonometric and exponential integrals.

Exam points

choose and apply integration by parts
evaluate integrals involving inverse trigonometric functions
use integration by parts in definite integrals and area problems

[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 $a x^{2} \tan ^{-1} x+b \int \frac{x^{2}}{1+x^{2}} \mathrm{~d} x$
Obtain $\frac{1}{2} x^{2} \tan ^{-1} x-\frac{1}{2} \int \frac{x^{2}}{1+x^{2}} \mathrm{~d} x$ , or equivalent
Obtain complete indefinite integral $\frac{1}{2}\left(x^{2} \tan ^{-1} x-x+\tan ^{-1} x\right)$ , or equivalent
Substitute limits having integrated twice
Obtain the given answer correctly
AG

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CAIE A-Level Mathematics A2 3.5.5 Inverse Tangent Questions question 1