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NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

The integral Ι = ∫ ( ( x ^ 2 ) + 2 x ^ 2 ( x ^ 2 ))dx = xH(x) + C,...

The integral

Ι = ∫ \left(\sin \left({x}^{2}\right) + 2{x}^{2}\cos \left({x}^{2}\right)\right)dx = xH\left(x\right) + C,

(where

C is the constant of integration). If the range of

H ( x )

is \left[a, b\right]

, then the value of

a + 2b

is equal to

Correct answer

1

Step-by-step solution

Given integral

Ι = ∫ \left(1 ⋅ \sin \left({x}^{2}\right) + x ⋅ 2xcos

\left({x}^{2}\right)\right)dx

<br />\left(∵ ∫ \left(f'g + f ⋅ {g}^{'}\right)dx = ∫ {\left(fg\right)}^{'}dx = f ⋅ g + C\right)

= x ⋅ \sin \left({x}^{2}\right) + C

∴ H\left(x\right) = \sin \left({x}^{2}\right),

whose range is \left[- 1,1\right]

<br />

∴ a = - 1, b = 1 ⇒ a + 2b = 1

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