Consider the following statements: (c or C is the constant of integration)( S1) ∫(x2+20)(xsinx+5cosx)2dx = −xcosx(xsinx+5cosx)+tanx+C( S2 ) ∫dx1+ecosx=1e2−1loge+cosx+e2−1sinx1+ecosx+c,(e is greater than 1) ( S3) 21−e2tan−1(1+e1−etanx2)+c,, (e lies between 0 and 1) (S4 ) If In=∫cotnxdx then I0+I1+2(I2+.....+I5)+I6+I7=−(u+u22+.....+u77)+C where u = cot x . Incorrect statement(s) is(are)

Statement 1. We can write I=∫x8(x2+20)(x5sinx+5x4cosx)2dx =∫(x5+20x3)cosx(x5sinx+5x4cosx)2(x5cosx)dx Note that ddx[x5 sin x+5x4cos x]=(x5+20x3)cos xIntegrating by parts, we get I=x5cos x(x5 sinx+5x4 cos x)+∫5x4cosx+x5sinx(x5 sinx+5x4 cos x)sec2 x dx =−xcosx(x sin x+5 cos x)+ tan x+C Statement 2, Statement 3. (t = tanx/2) If 0 1, Statement 4 In=∫cotnx dx= ∫cotn−2x(cosec2x−1)dx=−cotn−1xn−1−In−2+C1 ⇒ In+In−2=−cotn−1xn−1+C1Now, I0+I1+2(I2+....+I8)+I9+I10=(I2+I0)+(I3+I1)+(I4+I2)+(I5+I3)+(I6+I4)+(I7+I5)+(I8+I6)+(I9+I7)+(I10+I8) =−(cotx1+cot2x2+....+cot9x9)+C

Consider the following statements: (c or C is the constant of integration)( S1) ∫(x2+20)(xsinx+5cosx)2dx = −xcosx(xsinx+5cosx)+tanx+C( S2 ) ∫dx1+ecosx=1e2−1loge+cosx+e2−1sinx1+ecosx+c,(e is greater than 1) ( S3) 21−e2tan−1(1+e1−etanx2)+c,, (e lies between 0 and 1) (S4 ) If In=∫cotnxdx then I0+I1+2(I2+.....+I5)+I6+I7=−(u+u22+.....+u77)+C where u = cot x . Incorrect statement(s) is(are)

Statement 1. We can write I=∫x8(x2+20)(x5sinx+5x4cosx)2dx =∫(x5+20x3)cosx(x5sinx+5x4cosx)2(x5cosx)dx Note that ddx[x5 sin x+5x4cos x]=(x5+20x3)cos xIntegrating by parts, we get I=x5cos x(x5 sinx+5x4 cos x)+∫5x4cosx+x5sinx(x5 sinx+5x4 cos x)sec2 x dx =−xcosx(x sin x+5 cos x)+ tan x+C Statement 2, Statement 3. (t = tanx/2) If 0 1, Statement 4 In=∫cotnx dx= ∫cotn−2x(cosec2x−1)dx=−cotn−1xn−1−In−2+C1 ⇒ In+In−2=−cotn−1xn−1+C1Now, I0+I1+2(I2+....+I8)+I9+I10=(I2+I0)+(I3+I1)+(I4+I2)+(I5+I3)+(I6+I4)+(I7+I5)+(I8+I6)+(I9+I7)+(I10+I8) =−(cotx1+cot2x2+....+cot9x9)+C

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Consider the following statements: (c or C is the constant of integration)
( $S_{1}$ ) $\int \frac{(x^{2} + 20)}{{(x \sin x + 5 \cos x)}^{2}} d x$ = $- \frac{x}{\cos x (x \sin x + 5 \cos x)} + \tan x + C$
( $S_{2}$ ) $\int \frac{d x}{1 + e \cos x} = \frac{1}{\sqrt{e^{2} - 1}} \log \frac{e + \cos x + \sqrt{e^{2} - 1} \sin x}{1 + e \cos x} + c$ ,(e is greater than 1)
( $S_{3}$ ) $\frac{2}{\sqrt{1 - e^{2}}} \tan^{- 1} (\sqrt{\frac{1 + e}{1 - e}} \tan \frac{x}{2}) + c,$ , (e lies between 0 and 1)
( $S_{4}$ ) If $I_{n} = \int \cot^{n} x d x$ then $I_{0} + I_{1} + 2 (I_{2} + ..... + I_{5}) + I_{6} + I_{7} = - (u + \frac{u^{2}}{2} + ..... + \frac{u^{7}}{7}) + C$ where u = cot x .
Incorrect statement(s) is(are)

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$S_{4}$

$S_{2}$

$S_{3}$

$S_{1}$

answer is C, D.

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Detailed Solution

Statement 1. We can write
$I = \int \frac{x^{8} (x^{2} + 20)}{{(x^{5} \sin x + 5 x^{4} \cos x)}^{2}} d x$
$= \int \frac{(x^{5} + 20 x^{3}) \cos x}{{(x^{5} \sin x + 5 x^{4} \cos x)}^{2}} (\frac{x^{5}}{\cos x}) d x$
Note that
$\frac{d}{d x} [x^{5} \sin x + 5 x^{4} \cos x] = (x^{5} + 20 x^{3}) \cos x$
Integrating by parts, we get
   $I = \frac{x^{5}}{\cos x (x^{5} \sin x + 5 x^{4} \cos x)} + \int \frac{5 x^{4} \cos x + x^{5} \sin x}{(x^{5} \sin x + 5 x^{4} \cos x)} \sec^{2} x d x$
   $= - \frac{x}{\cos x (x \sin x + 5 \cos x)} + \tan x + C$
Statement 2, Statement 3.
Question Image (t = tanx/2)
If 0 < e < 1, ,
If e > 1,
Statement 4
$I_{n} = \int \cot^{n} x d x = \int \cot^{n - 2} x (\cos e c^{2} x - 1) d x = \frac{- \cot^{n - 1} x}{n - 1} - I_{n - 2} + C_{1}$
$\Rightarrow I_{n} + I_{n - 2} = \frac{- \cot^{n - 1} x}{n - 1} + C_{1}$
Now, $I_{0} + I_{1} + 2 (I_{2} + .... + I_{8}) + I_{9} + I_{10}$
$= (I_{2} + I_{0}) + (I_{3} + I_{1}) + (I_{4} + I_{2}) + (I_{5} + I_{3}) + (I_{6} + I_{4}) + (I_{7} + I_{5}) + (I_{8} + I_{6}) + (I_{9} + I_{7}) + (I_{10} + I_{8})$
$= - (\frac{\cot x}{1} + \frac{\cot^{2} x}{2} + .... + \frac{\cot^{9} x}{9}) + C$