Advertisements
Advertisements
Question
`int sin(logx) "d"x`
Advertisements
Solution
Let I = `int sin(log x) "d"x`
Put log x = t
∴ x = et
∴ dx = et dt
∴ I = `int sin "t" * "e"^"t" "dt"`
= `sin "t" int "e"^"t" "dt" - int ["d"/"dt" (sin "t") int "e"^"t" "dt"]"dt"`
= `sin "t"* "e"^"t" - int cos "t" * "e"^"t" "dt"`
= `"e"^"t" sin "t" - [cos "t" int "e"^"t" "dt" - int ("d"/"dt"(cos "t") int "e"^"t" "dt")"dt"]`
= `"e"^"t" sin "t" - ["e"^"t" cos "t" - int(- sin "t")"e"^"t" "dt"]`
= `"e"^"t" sin "t" - "e"^"t"cos "t" - int sin "t" * "e"^"t" "dt"`
∴ I = `"e"^"t"(sin "t"- cos "t") - "I" + "c"_1`
∴ 2I = `"e"^"t"(sin "t" - cos "t") + "c"_1`
∴ I = `"e"^"t"/2 (sin "t" - cos "t") + "c"_1/2`
∴ I = `x/2 [sin (log x) - cos(log x)] + "c"`,
where c = `"c"_1/2`
RELATED QUESTIONS
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int 1/(4x^2 - 20x + 17) "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
`int 1/(4x^2 - 20x + 17) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
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 (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
