What is ∫ In (x2 ) dx equal to?

By BYJU'S Exam Prep

Updated on: October 17th, 2023

(I) 2x In (x) – 2x + c

(II) 2/x + c

(II) 2x ln (x) + c

(IV) 2 ln(x)/x – 2x + c

∫ In (x2 ) dx equal to 2x ln x – 2x + c

∫u . v dx = u . ∫v dx – ∫ du/dx . ∫v dx

ILATE Rule: I – Inverse function, L – Logarithmic function, A- Algebraic function, T – Trigonometric function & E-Exponential function.

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We have to find the value of ∫ In (x2 ) dx. Let x2 = u

⇒ x = √u

⇒ dx = 1/(2√u) du

On putting the value of x and dx in the given integral

∫ In (x2 ) = ∫ In u . 1/(2√u) du

= ½ ∫ In u . 1/√u du

= ½ [ln u . ∫1/ √u du – ∫1/u . ∫ 1/ √u du]

= ½ [ln u . 2√u – ∫1/u . 2 √u du]

= ½ [2√u ln u – 4√u] + c

Now, On putting u = x2 , we get

= ½ [2x ln x2 – 4c] + c

= x ln x2 – 2x + c

= 2x ln x – 2x + c

Summary:

What is ∫ In (x2 ) dx equal to? (A) 2x In (x) – 2x + c (B) 2/x + c (C) 2x ln (x) + c (D) 2 ln(x)/x – 2x + c

∫ In (x2 ) dx equal to 2x ln x – 2x + c.