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∫√(x) ln x dx is equal to Single correct option
∫ x ln x d x \displaystyle\int\sqrt{x}\,\ln x\,dx ∫ x ln x d x is equal to
A. 2 3 x 3 / 2 ln x − 2 3 x 3 / 2 + C \frac23x^{3/2}\ln x-\frac23x^{3/2}+C 3 2 x 3/2 ln x − 3 2 x 3/2 + C B. 2 3 x 3 / 2 ln x + 4 9 x 3 / 2 + C \frac23x^{3/2}\ln x+\frac49x^{3/2}+C 3 2 x 3/2 ln x + 9 4 x 3/2 + C C. 3 2 x 3 / 2 ln x − 9 4 x 3 / 2 + C \frac32x^{3/2}\ln x-\frac94x^{3/2}+C 2 3 x 3/2 ln x − 4 9 x 3/2 + C D. 2 3 x 3 / 2 ln x − 4 9 x 3 / 2 + C \frac23x^{3/2}\ln x-\frac49x^{3/2}+C 3 2 x 3/2 ln x − 9 4 x 3/2 + C Show answer and solution Answer: (D) 2 3 x 3 / 2 ln x − 4 9 x 3 / 2 + C \frac23x^{3/2}\ln x-\frac49x^{3/2}+C 3 2 x 3/2 ln x − 9 4 x 3/2 + C
By parts with
u = ln x , d v = x 1 / 2 d x u=\ln x,\ dv=x^{1/2}dx u = ln x , d v = x 1/2 d x :
2 3 x 3 / 2 ln x − ∫ 2 3 x 3 / 2 ⋅ 1 x d x = 2 3 x 3 / 2 ln x − 2 3 ⋅ 2 3 x 3 / 2 \frac23x^{3/2}\ln x-\int\frac23x^{3/2}\cdot\frac1x\,dx=\frac23x^{3/2}\ln x-\frac23\cdot\frac23x^{3/2} 3 2 x 3/2 ln x − ∫ 3 2 x 3/2 ⋅ x 1 d x = 3 2 x 3/2 ln x − 3 2 ⋅ 3 2 x 3/2 , i.e. the second term is
4 9 x 3 / 2 \frac49x^{3/2} 9 4 x 3/2 , not
2 3 \frac23 3 2 . Differentiating the answer returns
x ln x \sqrt x\ln x x ln x . The other options slip on the sign or the reciprocal of
3 2 \frac32 2 3 .
Original question written in the NDA 2017 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake