Integration (3)

Calculus Level pending

π ÷ ∫ 0 ∞ t ( t + 1 ) ( t 2 + 1 ) d t = ? \large\pi \div \int_0^\infty \dfrac t{(t+1)(t^2+1)} \, dt = \, ?


The answer is 4.

Rishabh Deep Singh by Rishabh Deep Singh , 23, India

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1 solution

Tom Engelsman
Apr 26, 2020

The integral evaluates according to:

∫ 0 ∞ t + 1 2 ( t 2 + 1 ) − 1 2 ( t + 1 ) d t ⇒ 1 2 arctan ⁡ ( t ) − 1 2 ln ⁡ ( t + 1 ) + 1 4 ln ⁡ ( t 2 + 1 ) ∣ 0 ∞ = π 4 . \int_{0}^{\infty} \frac{t+1}{2(t^2+1)} - \frac{1}{2(t+1)} dt \Rightarrow \frac{1}{2} \arctan(t) - \frac{1}{2} \ln(t+1) + \frac{1}{4} \ln(t^2 + 1) |_{0}^{\infty} = \boxed{\frac{\pi}{4}}.

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