If the integral Ι = ∫ x^ x ( x ⋅ ln x + ( x)/(x))dx = (f (x))g (x) + c( x > 0) , then the range of y = g (x) is (where, c & is an arbitrary constant)

Correct answer: A. [- 1,1] Let x^ sin x = t; then, I = ∫ dt = t + c = x^ sin x + c Hence, range of x ∈ [ - 1, 1 ]

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If the integral Ι = ∫ x^ x ( x ⋅ ln x + ( x)/(x))dx = (f (x))g (x)...

If the integral $Ι = ∫ x^{\sin x}\left(\cos x ⋅ ln x + \dfrac{\sin x}{x}\right)dx = \left(f \left(x\right)\right)g \left(x\right) + c\left(\forall x > 0\right)$ , then the range of $y = g \left(x\right)$ is (where, $c$ & is an arbitrary constant)

Options

A $\left[- 1,1\right]$
B $\left[0,1\right]$
C $\left[0,1\right)$
D $\left(- 1,1\right)$

Correct answer

A. $\left[- 1,1\right]$

Step-by-step solution

Let $x^{sin x} = t;$ then,

$I = ∫ dt = t + c = x^{sin x} + c$

Hence, range of $\sin x \in [ - 1, 1 ]$

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