Comparing integral

Calculus Level 3

∫ 0 1 1 1 + x π 2 d x \large{\displaystyle \int^{1}_{0} \dfrac{1}{1+x^{\frac{\pi}{2}}} \, dx}

Let I I denote the value of the integral above. Which of these answer choices is correct?

ln ⁡ ( 2 ) < I < π 4 \ln(2) < I < \frac\pi4 I < ln ⁡ ( 2 ) I < \ln(2) None of these choices I < ln ⁡ ( 2 ) , I > π 4 I < \ln(2), I > \frac\pi4 I > π 4 I > \frac\pi4

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Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

Tanishq Varshney
Jun 24, 2015

1 1 + x < 1 1 + x π 2 < 1 1 + x 2 ∀ x ∈ ( 0 , 1 ) \huge{\frac{1}{1+x}<\frac{1}{1+x^{\frac{\pi}{2}}}<\frac{1}{1+x^{2}} \quad \forall x\in (0,1)}

integrating from 0 0 to 1 1

l n ( 2 ) < I < π 4 \large{ln(2)<I<\frac{\pi}{4}}

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