Advertisements
Advertisements
प्रश्न
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Advertisements
उत्तर
Let I = `int _0^(pi/4) "log"(1+"tan""x")"dx"`
= `int _0^(pi/4) "log"(1+ "tan""x")"dx"`
`=int _0^(pi/4) "log"{1+"tan"(pi/4-"x")} "dx"`
`(because int _0^"a" "f" ("x") "dx" int "f"("a" -"x")"dx")`
`=int _0^(pi/4)"log"{1+(("tan"pi/4 - "tan""x"))/(1+"tan"pi/4"tan""x")} "dx"`
`=int _0^(pi/4) "log"{1+(1-"tan""x")/(1+ "tan""x")} "dx"`
`=int _0^(pi/4) "log"{(1 + "tan""x" +1 -"tan""x")/(1 + "tan""x")}"dx"`
`=int _0^(pi/4) "log"(2/(1+"tan""x")) "dx"`
`=int _0^(pi/4) {"log" 2 -"log"(1+ "tan""x")} "dx"`
`=int _0^(pi/4) "log"2"dx" - int _0^(pi/4) "log" (1+"tan""x")"dx"`
`"I" = "log"2["x"]int _0^(pi/4) - "I"`
2I = `"log" 2 [pi/4-0]`
`"I" = pi/8 ."log"2`
` therefore int _0^(pi/4) "log"(1 +"tan""x")"dx" = pi/8"log"2`
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`sin x/(1+ cos x)`
Write a value of
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
`int "dx"/(9"x"^2 + 1)= ______. `
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
Evaluate `int (5"x" + 1)^(4/9)` dx
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate `int(1 + x + x^2/(2!) )dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
What is integration by substitution?
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
After putting \[t=\cos x\], which integral is obtained from \[\int\sin^2x\cos^2x(\sin x)\,dx\]?
