The correct option is C π4
I=a∫0dxx+√a2−x2
Putting
x=asinθ⇒dx=acosθ dθ
When
x=0=asinθ⇒θ=0x=a=asinθ⇒θ=π2
So, the integral become,
⇒I=π/2∫0cosθdθsinθ+cosθ ⋯(1)
Using property,
⇒I=π/2∫0cos(π2−θ)dθsin(π2−θ)+cos(π2−θ)⇒I=π/2∫0sinθdθcosθ+sinθ ⋯(2)
Adding (1) and (2),
⇒2I=π/2∫0sinθ+cosθsinθ+cosθdθ⇒I=12π/2∫01 dθ⇒I=π4