Integrate the following functions w.r.t. x : 4x(4x+4) - Mathematics and Statistics

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Sum

Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`

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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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Chapter 3: Indefinite Integration - Exercise 3.3 [Page 138]

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