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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If a function f: R → R is defined as f(x) = ∫ x^ 8 + 4 x^ 4 - 2x^ 2...

If a function f:R→Rf: R \to R is defined as f(x)=∫x8+4x4−2x2+2dxf\left(x\right) = ∫ \dfrac{x^{8} + 4}{x^{4} - 2x^{2} + 2}dx and f(0)=1f\left(0\right) = 1, then which of the following is correct?

Options

  1. Af(x)isanevenfunctionf \left(x\right) is an even function
  2. Bf(x)isanontofunctionf \left(x\right) is an onto function
  3. Cf(x)isanoddfunctionf \left(x\right) is an odd function
  4. Df(x)ismanyonefunctionf \left(x\right) is many one function

Correct answer

B. f(x)isanontofunctionf \left(x\right) is an onto function

Step-by-step solution

f(x)=∫(x8+4+4x4)−(4x4)x4−2x2+2dxf\left(x\right) = ∫ \dfrac{\left(x^{8} + 4 + 4x^{4}\right) - \left(4x^{4}\right)}{x^{4} - 2x^{2} + 2}dx
=∫((x4+2)2−(2x2)2x4−2x2+2)dx= ∫ \left(\dfrac{{\left(x^{4} + 2\right)}^{2} - {\left(2x^{2}\right)}^{2}}{x^{4} - 2x^{2} + 2}\right)dx
=∫(x4+2x2+2)(x4−2x2+2)(x4−2x2+2)dx= ∫ \left(x^{4} + 2x^{2} + 2\right)\left(x^{4} - 2x^{2} + 2\right)\left(x^{4} - 2x^{2} + 2\right)dx
f(x)=x55+2x33+2x+Cf\left(x\right) = \dfrac{x^{5}}{5} + \dfrac{2x^{3}}{3} + 2x + C
f(0)=0+0+0+C=1⇒C=1f\left(0\right) = 0 + 0 + 0 + C = 1 ⇒ C = 1
f(x)=x55+2x33+2x+1f \left(x\right) = \dfrac{x^{5}}{5} + \dfrac{2x^{3}}{3} +2x + 1
Range of f(x)f\left(x\right) is RR, so f(x)f ( x ) is an onto function
f(−x)f \left(-x\right)\ne f \left(x\right),sonoteven<br/>, so not even<br />f \left(-x\right) \ne - f \left(x\right), so not odd
f′f^{'}\left(x\right)>>0 \forall x \in \mathbb{R}, so f(x)f ( x )& is one-one
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