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Question
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Solution
Let I = `int sqrt(4^x(4^x + 4)).dx`
= `int 2^xsqrt((2^x)^2 + 2^2).dx`
Put 2x = t
∴ 2x log 2 dx = dt
∴ 2x dx = `(1)/log2.dt`
∴ I = `int sqrt(t^2 + 2^2). dt/log2`
= `(1)/log2 int sqrt(t^2 + 2^2).dt`
= `(1)/log2[t/2 sqrt(t^2 + 2^2) + 2^2/(2)log|t + sqrt(t^2 + 2^2)|] + c`
= `(1)/log2 [2^x/2 sqrt(4^x + 4) + 2log|2^x + sqrt(4^x + 4)|] + c`
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