The integral I = ∫ ((1)/(x ⋅ sec x) - ln (x^ sin x ))dx simplifies to (where, c is the constant of integration)

Correct answer: B. (ln x)(cos x) + c Given I = ∫ (cos x ⋅ (1)/(x) + (- sin x)(ln x))dx Since, ∫ (f ⋅ g^ ' + f^ ' (g))dx = f ⋅ g + c ⇒ I = (cos x)(ln x) + c

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

The integral I = ∫ ((1)/(x ⋅ sec x) - ln (x^ sin x ))dx simplifies...

The integral

I = ∫

\left(\dfrac{1}{x ⋅ sec x} - ln

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

simplifies to (where,

c

is the constant of integration)

Options

A $\left(ln x\right)\left(sin x\right) + c$
B $\left(ln x\right)\left(cos x\right) + c$
C $xln \left(sin x\right) + c$
D $xln \left(cos x\right) + c$

Correct answer

B. $\left(ln x\right)\left(cos x\right) + c$

Step-by-step solution

Given

I = ∫ \left(cos x ⋅ \dfrac{1}{x} + \left(- sin x\right)\left(ln x\right)\right)dx

Since,

∫ \left(f ⋅ g^{'} + f^{'}\left(g\right)\right)dx

= f ⋅ g + c

⇒ I = \left(cos x\right)\left(ln x\right) + c

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