Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Advertisements
उत्तर
Let I = `int_"0"^pi (x"d"x)/(1 + sin x)` .....(i)
= `int_0^pi (pi - x)/(1 + sin(pi - x)) "d"x` ......`["Using" int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x)"d"x]`
= `int_0^pi (pi - x)/(1 + sinx) "d"x` ......(ii)
Adding (i) and (ii), we get
2I = `int_0^pi (x/(1 + sinx) + (pi - x)/(1 + sinx)) "d"x`
= `int_0^pi ((x + pi - x)/(1 + sinx))"d"x`
= `int_0^pi pi/(1 + sin x) "d"x`
= `pi int_0^pi 1/(1 + sinx) "d"x`
= `pi int_0^pi (1.(1 - sinx))/((1 + sinx)(1 - sinx)) "d"x`
= `pi int_0^pi (1 - sinx)/(1 - sin^2x) "d"x`
= `pi int_0^pi (1 - sinx)/(cos^x) "d"x`
= `pi int_0^pi (1/(cos^2x) - sinx/(cos^2x))"d"x`
= `pi int_0^pi (sec^2x - secx tanx)"d"x`
= `pi[tanx - sec]_0^pi`
= `pi[tan pi - tan 0) - (sec pi - sec 0)]`
2I = `pi[0 - (-1 - 1)`
= `pi`(2)
∴ I = `pi`
Hence, I = `pi`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Find: `I=intdx/(sinx+sin2x)`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (xdx)/((x - 1)(x - 2))` equals:
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int 1/(4x^2 - 20x + 17) "d"x`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x log x "d"x`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
`int 1/(x^2 + 1)^2 dx` = ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
