Integration 5

Calculus Level 2

∫ 0 1 2 d x 1 − 3 x 2 \large \int^{\frac 12}_0 \dfrac{dx}{\sqrt{1-3x^2}}

Find the value of above integral to 4 decimal places.

Rajshahi University (2017-2018)-C unit]

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The answer is 0.6046.

Nazmus Sakib by Nazmus sakib , 24, Bangladesh

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1 solution

Chew-Seong Cheong
Oct 31, 2017

I = ∫ 0 1 2 1 1 − 3 x 2 d x Let sin ⁡ θ = 3 x ⟹ cos ⁡ θ d θ = 3 d x = 1 3 ∫ 0 π 3 cos ⁡ θ cos ⁡ θ d θ = 1 3 θ ∣ 0 π 3 = π 3 3 = 0.6046 \begin{aligned} I & = \int_0^\frac 12 \frac 1{\sqrt{1-3x^2}} dx & \small \color{#3D99F6} \text{Let }\sin \theta = \sqrt 3 x \implies \cos \theta \ d \theta = \sqrt 3 \ dx \\ & = \frac 1{\sqrt 3} \int_0^\frac \pi 3 \frac {\cos \theta}{\cos \theta} d \theta \\ & = \frac 1{\sqrt 3} \theta \ \bigg|_0^\frac \pi 3 \\ & = \frac \pi {3\sqrt 3} \\ & = \boxed{0.6046} \end{aligned}

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