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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If the integral I = ∫ x^ 5 1 + x^ 3 dx = K x^ 3 + 1 (x^ 3 - 2) + C,...

If the integral I=∫x51+x3dx=Kx3+1(x3−2)+C,I = ∫ \dfrac{x^{5}}{\sqrt{1 + x^{3}}}dx = K\sqrt{x^{3} + 1}\left(x^{3} - 2\right) + C, (where, CC is the constant of integration), then the value of 9K9K& is equal to

Options

  1. A44
  2. B22
  3. C66
  4. D1010

Correct answer

B. 22

Step-by-step solution

Let 1+x3=t21 + x^{3} = t^{2}
⇒3x2dx=2tdt⇒ 3x^{2}dx = 2tdt
⇒I=∫(t2−1)2tdt3t⇒ I = ∫ \dfrac{\left(t^{2} - 1\right)2tdt}{3t}
⇒I=23∫(t2−1)dt⇒ I = \dfrac{2}{3}∫ \left(t^{2} - 1\right)dt
=2t39−23t+C= \dfrac{2t^{3}}{9} - \dfrac{2}{3}t + C
=29t(t2−3)+C= \dfrac{2}{9}t\left(t^{2} - 3\right) + C
=291+x3(x3−2)+C= \dfrac{2}{9}\sqrt{1 + x^{3}}\left(x^{3} - 2\right) + C
∴K=29⇒9K=2∴ K = \dfrac{2}{9} ⇒ 9K = 2
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