CBSE Class 12-science Answered

Put x = tan^-1(2t/(1-t²)) or tanx = 2t/(1-t²)

then dx = 2 dt/(1+t²) and x = 0, t = 0; x = π/2, t = ±1

and sinx = 2t/(1+t²) and cosx = (1-t²)/(1+t²)

2cosx + 4sinx = 2((1-t²)/(1+t²) + 4(2t/(1+t²)) = [2(1-t²) + 8t]/(1+t²) = 2[(1-t²) + 4t]/(1+t²)

dx/(2cosx + 4sinx) = dt/{(1+t²)[(1-t²) + 4t]/(1+t²)} = dt/[(1-t²) + 4t] = dt/[5 - (t-2)²]

This can further be solved by partial fraction or using standard integral, and subsequent use of limits will complete the proof.

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