Question

Integrate the following function

$\int \: \frac{tanx}{ {a}^{2} + {b}^{2} {tan}^{2} x} \: dx$

Please explain step by step.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm :\longmapsto\:\displaystyle\int\rm \frac{tanx}{ {a}^{2} + {b}^{2} {tan}^{2} x} \: dx$

can be rewritten as

$\rm \: = \: \displaystyle\int\rm \frac{ \dfrac{sinx}{cosx} }{ {a}^{2} + {b}^{2} \dfrac{ {sin}^{2} x}{ {cos}^{2} x}} dx$

$\rm \: = \: \displaystyle\int\rm \frac{ \dfrac{sinx}{cosx} }{ \dfrac{ {a}^{2} {cos}^{2}x + {b}^{2} {sin}^{2} x}{ {cos}^{2} x}}dx \:$

$\rm \: = \: \displaystyle\int\rm \frac{sinx \: cosx}{ {a}^{2} {cos}^{2}x + {b}^{2} {sin}^{2}x} \: dx$

Now, to evaluate this integral further, we use Method of Substitution.

So, Substitute

$\purple{\rm :\longmapsto\: {a}^{2} {cos}^{2}x + {b}^{2} {sin}^{2}x = y}$

$\purple{\rm :\longmapsto\: [ - {a}^{2}(2cosx \: sinx) + {b}^{2}(2sinx \: cosx)] \: dx \: = \: dy}$

$\purple{\rm :\longmapsto\:sinxcosx[ {2b}^{2} - {2a}^{2}] \: dx \: = \: dy}$

$\purple{\rm :\longmapsto\:sinxcosx \: dx \: = \: \dfrac{dy}{2( {b}^{2} - {a}^{2})} }$

So, on substituting all these values, we get

$\rm \: = \: \dfrac{1}{2( {b}^{2} - {a}^{2})} \displaystyle\int\rm \frac{dy}{y}$

$\rm \: = \: \dfrac{1}{2( {b}^{2} - {a}^{2})} log |y| + c$

$\rm \: = \: \dfrac{1}{2( {b}^{2} - {a}^{2})} log \bigg| {a}^{2} {cos}^{2}x + {b}^{2} {sin}^{2}x\bigg| + c$

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Formula Used

$\boxed{\tt{ \dfrac{d}{dx}sinx = cosx}}$

$\boxed{\tt{ \dfrac{d}{dx}cosx = - sinx}}$

$\boxed{\tt{ \dfrac{d}{dx} {x}^{n} = {nx}^{n - 1} }}$

$\boxed{\tt{ \int \: \frac{1}{x} \: dx \: = \: log |x| + c \: }}$

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LEARN MORE

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$