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Statement I - The value of the integral $integral subscript straight pi divided by 6 end subscript superscript straight pi divided by 3 end superscript fraction numerator dx over denominator 1 plus space square root of tan space straight x end fraction$ is equal to π/6.
Statement II- $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript straight a superscript straight b straight f left parenthesis straight a plus straight b minus straight x right parenthesis space dx$

Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I
Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I
Statement -I is True; Statement -II is False.
Statement -I is True; Statement -II is False.

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12.
If the integral $integral fraction numerator 5 space tan space straight x over denominator tan space straight x minus 2 end fraction straight d space straight x space equals space straight x space plus space straight a space log space vertical line space sin space straight x minus space 2 space cos space straight x vertical line space plus space straight k comma$ then an equal to

-1

-2

1

1

integral fraction numerator 5 space tan space straight x over denominator tan space straight x minus 2 end fraction straight d space straight x space equals space straight x space plus space straight a space log space vertical line space sin space straight x minus space 2 space cos space straight x vertical line space plus space straight k comma space... space left parenthesis straight i right parenthesis

Now, let us assume that I,

straight I space equals integral fraction numerator 5 space tan space straight x over denominator tan space straight x minus 2 end fraction straight d space straight x

Multiplying by cos x in numerator and denominator, we get

straight I space equals space integral fraction numerator 5 space sin space straight x over denominator sin space straight x minus 2 space cos space straight x end fraction straight d space straight x

This special integration requires special substitution of type
N' = A (D') +

straight B space open parentheses fraction numerator dD apostrophe over denominator dx end fraction close parentheses

⇒ Let 5 sin x = (A + 2B) sin x + (B- 2A) cos x
Comapring the coefficients of sin x and cosx,
we get
A +2B =5 and B- 2A = 0
Solving the above two equations in A and B
we get
A = 1 an B= 2
⇒ 5 sin x = ( sin x - 2 cos x)+ 2 (cos x + 2 sin x)

rightwards double arrow space straight I space equals space integral fraction numerator 5 space sin space straight x over denominator sin space straight x minus space 2 space cos space straight x end fraction dx space equals space integral fraction numerator sin space straight x minus space 2 space cos space straight x right parenthesis space plus space 2 space left parenthesis cos space straight x space plus 2 space sin space straight x right parenthesis space dx over denominator left parenthesis sin space straight x minus 2 space cos space straight x right parenthesis end fraction rightwards double arrow space straight I space equals space integral fraction numerator left parenthesis sin space straight x minus 2 space cos space straight x right parenthesis plus space 2 space left parenthesis cos space straight x space plus 2 space sin space straight x right parenthesis over denominator sin space straight x minus space 2 cos space straight x end fraction space dx rightwards double arrow space straight I space equals space integral fraction numerator sin space straight x minus space 2 space cos space straight x over denominator sin space straight x minus space 2 space cos space straight x end fraction dx space plus space 2 space integral fraction numerator left parenthesis cos space straight x space plus 2 space sin space straight x right parenthesis over denominator left parenthesis sin space straight x minus 2 space cos space straight x right parenthesis end fraction dx rightwards double arrow space straight I space equals integral space 1 space dx space plus space 2 space integral fraction numerator straight d left parenthesis sin space straight x minus 2 space cos space straight x right parenthesis over denominator left parenthesis sin space straight x minus space 2 space cos space straight x right parenthesis space end fraction rightwards double arrow space straight I space space equals space straight x space plus 2 space log space vertical line sin space straight x minus space 2 space cos space straight x right parenthesis vertical line space plus straight k space... space left parenthesis ii right parenthesis

where k is the constant of integration. Now, by comparing the value of l in eq. (i) and (ii) we get a = 2

⇒

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13.

If g(x) = $integral subscript 0 superscript straight x cos space 4 straight t space dt comma$ then g(x +π) equals

g(x)/g(π)
g(x) +g(π)
g (x) - g(π)
g (x) - g(π)

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14.

The value of $integral subscript 0 superscript 1 fraction numerator 18 space log space left parenthesis 1 plus straight x right parenthesis over denominator 1 plus straight x squared end fraction space dx$ is

$straight pi over 8 space log space 2$
$straight pi over 2 space log space 2$
log 2
log 2

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15.

The integral $integral subscript straight pi over 4 end subscript superscript fraction numerator 3 straight pi over denominator 4 end fraction end superscript fraction numerator dx over denominator 1 plus space cos space straight x end fraction$ is equal to

-1
-2
2
2

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16.

Let space straight I space space equals space integral subscript 0 superscript 1 space fraction numerator sin space straight x over denominator square root of straight x end fraction dx space and space straight J space equals space integral subscript 0 superscript 1 space fraction numerator cos space straight x over denominator square root of straight x end fraction space dx

then which one is true?

I > 2/ 3 and J > 2
I < 2/ 3 and J < 2
I < 2/ 3 and J > 2
I < 2/ 3 and J > 2

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17.

The value of $square root of 2 space integral fraction numerator sin space straight x space dx over denominator sin space open parentheses straight x minus begin display style straight pi over 4 end style close parentheses end fraction space is$

$space straight x plus space log open vertical bar cos space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus space straight C$
$straight x space minus log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c$
$straight x space plus space log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c$
$straight x space plus space log space open vertical bar sin space open parentheses straight x minus straight pi over 4 close parentheses close vertical bar space plus straight c$

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18.

let space straight F left parenthesis straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis space plus space straight f space open parentheses 1 over straight x close parentheses comma space where space straight f left parenthesis straight x right parenthesis space equals space integral subscript 1 superscript straight x fraction numerator log space straight t over denominator 1 plus straight t end fraction dt.

Then F(e) equals

90 Views

19.

The value of the integral, $integral subscript 3 superscript 6 fraction numerator square root of straight x over denominator square root of 9 minus straight x end root plus square root of straight x end fraction dx$ is

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20.

integral subscript 0 superscript straight pi space xf space left parenthesis sin space straight x space right parenthesis space dx

is equal to

$straight pi integral subscript 0 superscript straight pi space straight f left parenthesis cos space straight x right parenthesis space dx$
$straight pi integral subscript 0 superscript straight pi space straight f left parenthesis sin space straight x right parenthesis space dx$
$straight pi over 2 integral subscript 0 superscript straight pi space straight f left parenthesis cos space straight x right parenthesis space dx$
$straight pi over 2 integral subscript 0 superscript straight pi space straight f left parenthesis cos space straight x right parenthesis space dx$