JEE Main 2022 · 27 Jun · Shift 2 · Mathematics Q9

MathematicsSingle correct+4 / -1Question 9 of 90

The integral $\displaystyle\int_0^1\frac1{7^{\left[\frac1x\right]}}dx$ , where $[\cdot]$ denotes the greatest integer function, is equal to

1+6\log_e\left(\dfrac67\right)

1-6\log_e\left(\dfrac67\right)

\log_e\left(\dfrac76\right)

1-7\log_e\left(\dfrac67\right)

Answer and worked solution

Official answer: A

Split. On $\left(\frac1{n+1},\frac1n\right]$ , $\left[\frac1x\right]=n$ : $I=\displaystyle\sum_{n\ge1}7^{-n}\left(\frac1n-\frac1{n+1}\right)$ .

Sums. $\sum\frac{7^{-n}}n=\ln\frac76$ and $\sum\frac{7^{-n}}{n+1}=7\left(\ln\frac76-\frac17\right)$ .

Value. $I=1-6\ln\frac76=1+6\log_e\frac67$ .

(A)

Topics:Properties of definite integrals Special series sums

JEE Main 2022 · 27 Jun · Shift 2, Mathematics question 9