Question

integration of 2x^2+14x+21/√x^2+4x+1​

Answer 1

Step-by-step explanation:

I=∫dx(x2+4x+5)2

=∫dx[(x+2)2+1]2

Let x+2=tanθ,

then dx=sec2θdθ

I=∫sec2θdθ(tan2θ+1)2=∫sec2θdθsec4θ=12∫(1+cos2θ)dθ=12(θ+sinθcosθ)+C=12[tan−1(x+2)+x+2x2+4x+5]+c

Answer 2

Step-by-step explanation:

=∫dx(x2+4x+5)2

=∫dx[(x+2)2+1]2

Let x+2=tanθ,

then dx=sec2θdθ

I=∫sec2θdθ(tan2θ+1)2=∫sec2θdθsec4θ=12∫(1+cos2θ)dθ=12(θ+sinθcosθ)+C=12[tan−1(x+2)+x+2x2+4x+5]+c

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