First slide

Introduction to integration

Question

$If f (x) = \int \frac{(x^{2} + \sin^{2} x)}{1 + x^{2}} \sec^{2} x d x and f (0) = 0, then f (1) =$

Moderate

A

$1 - π / 4$
B

$π / 4 - 1$
C

$\tan 1 - π / 4$
D

$None of these$

Solution

$\begin{aligned} f (x) = \int \frac{(x^{2} + \sin^{2} x)}{1 + x^{2}} \sec^{2} x d x = \int \frac{x^{2} + (1 - \cos^{2} x)}{1 + x^{2}} \sec^{2} x d x \\ = \int (\sec^{2} x - \frac{1}{1 + x^{2}}) d x \\ = \tan x - \tan^{- 1} x + C \end{aligned}$
$\begin{array}{l} ∵ f (0) = 0, ∴ C = 0 \\ Thus, f (x) = \tan x - \tan^{- 1} x \end{array}$
$Hence, f (1) = \tan 1 - \tan^{- 1} 1 = \tan 1 - π / 4$

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