First slide

Introduction to integration

Question

$Evaluate \int (3 x^{2} \tan \frac{1}{x} - x \sec^{2} \frac{1}{x}) d x$

Moderate

A

$x^{3} \tan \frac{1}{x} + c$
B

$x^{2} \tan \frac{1}{x} + c$
C

$6 x \tan \frac{1}{x} + c$
D

none of these

Solution

$\begin{array}{l} = \int (3 x^{2} \tan \frac{1}{x} - x \sec^{2} \frac{1}{x}) d x \\ = \int 3 x^{2} \tan \frac{1}{x} d x - \int x \sec^{2} \frac{1}{x} d x (a p p l y i n t e g r a t i o n b y p a r t s o n l y f o r 1 s t i n t e g r a n d) \\ = \tan \frac{1}{x} x^{3} - \int (\sec^{2} \frac{1}{x}) (- \frac{1}{x^{2}}) x^{3} d x - \int x \sec^{2} \frac{1}{x} d x \\ = x^{3} \tan \frac{1}{x} + c \end{array}$

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