If I m -01 x m -ln x d x and J m -01 x m -ln x 2 d x where m - N- thenA- I 1- J 1-0B- 2 I 2-3 J 2-0C- I 3-2 J 3-0D- 2 I 4-5 J 4-0

The correct options are A I(1) + J(1) = 0 B 2I(2) + 3J(2) = 0 C I(3) + 2J(3) = 0 D 2I(4) + 5J(4) = 0I(m) = ∫^1_0 x^m. ln x dx = [ln x. x^m+1m+1]^1_0 1/m+1. ∫^ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Using Monotonicity to Find the Range of a Function

If I m =∫01 x...

Question

If $I (m) = \int_{0}^{1} x^{m} . ln x d x$ and $J (m) = \int_{0}^{1} x^{m} . (ln x)^{2} d x$ where $m \in N,$ then

I (1) + J (1) = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2 I (2) + 3 J (2) = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

I (3) + 2 J (3) = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2 I (4) + 5 J (4) = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct options are
A $I (1) + J (1) = 0$
B $2 I (2) + 3 J (2) = 0$
C $I (3) + 2 J (3) = 0$
D $2 I (4) + 5 J (4) = 0$
$I (m) = \int_{0}^{1} x^{m} . ln x d x = {[ln x . \frac{x^{m + 1}}{m + 1}]}_{0}^{1} - \frac{1}{m + 1} . \int_{0}^{1} x^{m} d x$
$= - \frac{1}{(m + 1)^{2}}$

$J (m) = \int_{0}^{1} x^{m} (ln x)^{2} d x$
$= {[\frac{x^{m + 1}}{m + 1} (ln x)^{2}]}_{0}^{2} - \frac{2}{m + 1} 1 \int 0 x^{m} ln x d x = 0 - \frac{2}{m + 1} I (m) = \frac{2}{(m + 1)^{3}}$

Suggest Corrections

Similar questions

Q. Suppose

I_{1} = \int_{0}^{\frac{π}{2}} c o s (π s i n^{2} x) d x, I_{2} = \int_{0}^{\frac{π}{2}} c o s (2 π s i n^{2} x) d x a n d I_{3} = \int_{0}^{\frac{π}{2}} c o s (π s i n x) d x,

then

Let f: R $\to$ R such that f(x + 2y) = f(x) + f(2y) + 4xy, $\forall$ x, y $ϵ$ R and f(0) = 0.

If $I_{1} = \int_{0}^{1} f (d x), I_{2} = \int_{- 1}^{0} f (x) d x, a n d I_{3} = \int_{- 1}^{1} f (x) d x, t h e n$

Q. If

^{n^{2} - n} C_{2} =^{n^{2} - n} C_{10}

, then

Q. If

I_{n} = \int {tan}^{n} x d x

then

I_{0} + I_{1} + 2 (I_{2} + I_{3} + . . . . + I_{8}) + I_{9} + I_{10}

equals

(n \in N)

Q. If

z = {(\frac{\sqrt{3}}{2} + \frac{i}{2})}^{5} + {(\frac{\sqrt{3}}{2} - \frac{i}{2})}^{5},

then
(a) Re (z) = 0
(b) Im (z) = 0
(c) Re (z) > 0, Im (z) > 0
(d) Re (z) > 0, Im (z) < 0

Join BYJU'S Learning Program

If I(m)=∫10xm.lnx dx and J(m)=∫10xm.(lnx)2dx where m∈N, then

The correct options are A I(1)+J(1)=0 B 2I(2)+3J(2)=0 C I(3)+2J(3)=0 D 2I(4)+5J(4)=0I(m)=∫10xm.lnx dx=[lnx.xm+1m+1]10−1m+1.∫10xmdx =−1(m+1)2 J(m)=∫10xm(lnx)2 dx =[xm+1m+1(lnx)2]10−2m+11∫0xmlnx dx=0−2m+1I(m)=2(m+1)3

If $I (m) = \int_{0}^{1} x^{m} . ln x d x$ and $J (m) = \int_{0}^{1} x^{m} . (ln x)^{2} d x$ where $m \in N,$ then