Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
Advertisements
Solution
Let I = `int (e^x.log(sin e^x))/tan(e^x).dx`
= `int log (sin e^x).e^x.cot (e^x) dx`
Put log (sin ex) = t
∴ `(1)/sin (e^x).cos(e^x).e^x dx` = dt
∴ ex . cot (ex) dx = dt
∴ I = `int t dt = t^2/(2) + c`
= `(1)/(2)[log (sine^x)]^2 + c`.
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
`int logx/(log ex)^2*dx` = ______.
`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?
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 (5"x" + 1)^(4/9)` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int (sin4x)/(cos 2x) "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int(1 - x)^(-2) dx` = ______.
Evaluate `int(3x^2 - 5)^2 "d"x`
`int(5x + 2)/(3x - 4) dx` = ______
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
`int secx/(secx - tanx)dx` equals ______.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int1/(x(x-1))dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
