Advertisements
Advertisements
Question
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Advertisements
Solution
Let I = `int_0^(pi/4) (dx)/(1 + tanx)`
= `int_0^(pi/4) (dx)/(1 + sinx/cosx)`
= `int_0^(pi/4) (cos x dx)/(cosx + sinx)`
= `1/2 int_0^(pi/4) (2cosx)/(cosx + sinx) dx`
= `1/2 int_0^(pi/4) (cosx + sinx + cosx - sinx)/(cosx + sinx) dx`
= `1/2 [int_0^(pi/4) (cosx + sinx)/(cosx + sinx) dx + int_0^(pi/4) (cosx - sinx)/(cosx + sinx) dx]`
= `1/2 [int_0^(pi/4) 1dx + int_0^(pi/4) (cosx - sinx)/(cosx + sinx) dx]`
= `1/2 (I_1 + I_2)`
Where, I1 = `int_0^(pi/4) 1dx`
= `[x]_0^(pi/4) = pi/4`
And I2 = `int_0^(pi/4) (cosx - sinx)/(cosx + sinx) dx`
Let cosx + sinx = t
⇒ (–sinx + cosx)dx = dt
When x = 0, t = 1
And x = `pi/4`, t = `2/sqrt(2)`
∴ I2 = `int_1^(2/sqrt(2)) (dt)/t`
= `[logt]_1^(2/sqrt(2))`
= `log 2/sqrt(2) - log 1`
= `log 2/sqrt(2) - 0`
= `log2^(3/2)`
= `3/2 log 2`
∴ I = `1/2(I_1 + I_2)`
or I = `1/2(pi/4 + 3/2 log 2)`
APPEARS IN
RELATED QUESTIONS
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Evaluate the following.
`int x^2 *e^(3x)`dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int (sinx)/(1 + sin x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int 1/sqrt(x^2 - a^2)dx` = ______.
`int(logx)^2dx` equals ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int e^(logcosx)dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
The value of `inta^x.e^x dx` equals
