Advertisements
Advertisements
Question
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Advertisements
Solution
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
RELATED QUESTIONS
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Write a value of
Write a value of\[\int \log_e x\ dx\].
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
`int logx/(log ex)^2*dx` = ______.
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int x^3"e"^(x^2) "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
`int 1/(sinx.cos^2x)dx` = ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
`int (logx)^2/x dx` = ______.
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int sin^2(x/2)dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
