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:
tan2(2x – 3)
Evaluate: `int 1/(x(x-1)) dx`
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate the following : `int (logx)2.dx`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate `int 1/(x (x - 1))` dx
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate `int 1/((2"x" + 3))` dx
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
`int x^x (1 + logx) "d"x`
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int dx/(1 + e^-x)` = ______
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
`int "cosec"^4x dx` = ______.
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate `int (1 + x + x^2/(2!)) dx`
