#11 of June 2018 Grade 12 - Natural Sciences CSAT (Korean SAT) Mock test

Calculus Level 2

Find the value of

∫ 1 2 x 3 x 2 − 1 d x . \int_1^{\sqrt{2}} x^3\sqrt{x^2-1}dx.

  • It is expected that you solve this in 2 minutes. (without calculator of course)

This problem is a part of <Grade 12 CSAT Mock test> series .

11 15 \dfrac{11}{15} 2 3 \dfrac{2}{3} 3 5 \dfrac{3}{5} 8 15 \dfrac{8}{15} 7 15 \dfrac{7}{15}

Boi (보이) by Boi (보이) , 19, South Korea

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1 solution

Boi (보이)
Jun 9, 2018

Let x 2 − 1 = t , x^2-1=t, so that 2 x d x = d t . 2x dx = dt. Then,

∫ 1 2 x 3 x 2 − 1 d x = 1 2 ∫ 0 1 ( t + 1 ) t d t = 1 2 [ 2 5 t 5 2 + 2 3 t 3 2 ] 0 1 = 8 15 . \int_1^{\sqrt{2}}x^3\sqrt{x^2-1}dx \\ =\dfrac{1}{2}\int_0^1 (t+1)\sqrt{t}dt \\ =\dfrac{1}{2}\left[\frac{2}{5}t^{\frac{5}{2}}+\frac{2}{3}t^{\frac{3}{2}}\right]_0^1 \\ =\boxed{\dfrac{8}{15}}.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Great question! I first thought that substitution using Secx would give some fruit. But got entangled in the modulus. As I think it would have been the easiest prob on your exam!

Shivam Hinduja - 2 years, 12 months ago

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