First slide

Introduction to integration

Question

$\int x^{3} d (\tan^{- 1} x) is equal to$

Easy

A

$\frac{x^{2}}{2} + \frac{1}{2} \ln (1 + x^{2}) + C$
B

$- \frac{x^{2}}{2} + \frac{1}{2} \ln (1 + x^{2}) + C$
C

$- \frac{x^{2}}{2} - \frac{1}{2} \ln (1 + x^{2}) + C$
D

$\frac{x^{2}}{2} - \frac{1}{2} \ln (1 + x^{2}) + C$

Solution

$\begin{array}{l} I = \int x^{3} d (\tan^{- 1} x) = \int \frac{x^{3}}{1 + x^{2}} dx \\ = \int (x - \frac{x}{1 + x^{2}}) dx = \frac{x^{2}}{2} - \frac{1}{2} \ln (1 + x^{2}) + C \end{array}$

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