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# Evaluate : $\int \large\frac{\sec^2x}{3+\tan\:x}$$dx Can you answer this question? ## 1 Answer 0 votes Toolbox: • Method of substitution : • Given f(x)dx can be transformed into another form by changing independent variable x to t by substituting x=g(t). • Consider I=\int f(x)dx. • Put x=g(t) so that \large\frac{dx}{dt}$$=g'(t).$
• $\Rightarrow dx=g'(t)dt.$
• Thus $I=\int f(g(t).g'(t))dt.$
• $\large\frac{d}{dx}$$(\tan x)=\sec^2xdx Step 1: Let I=\int \large\frac{\sec^2 x}{3+\tan x}$$dx$
Let $\tan x=t$
Differentiating with respect to $x$ we get,
$\sec^2 xdx=dt$
Step 2:
$\therefore I=\int\large\frac{dt}{3+t}$
$\quad\;\;=\log \mid 3+t\mid+c$
Substituting for $t$ we get,
$I=\log\mid 3+\tan x\mid+c$