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Question
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
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Solution
Let I = `int 2^x/(4^x - 3*2^x - 4)*dx`
= `int 2^x/((2^x)^2 - 3*2^x - 4)`
Put 2x = t
∴ 2x log 2 dx = dt
∴ 2x dx = `(1)/log2*dt`
∴ I = `(1)/log2 int dt/(t^2 - 3t - 4)`
= `(1)/log2 int (1)/((t + 1)(t - 4))*dt`
= `(1)/(5log2) int ((t + 1) - (t - 4))/((t - 4)(t - 4))*dt` ...[Note this step.]
= `(1)/(5log2) int [1/(t - 4) - 1/(t + 1)]*dt`
= `(1)/(5log2) [int 1/(t - 4)*dt - int 1/(t + 1)*dt]`
= `(1)/(5log2)[log|t - 4| - log|t + 1|] + c`
= `(1)/(5log2)log|(2^x - 4)/(2^x + 1)| + c`.
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