Advertisements
Advertisements
Question
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Advertisements
Solution
Let I = `int ("e"^x*log(sin"e"^x))/(tan("e"^x)) "d"x`
= `int"e"^x*log(sin"e"^x)*cot("e"^x) "d"x`
Put log(sin ex) = t
Differentiating w.r.t. x, we get
`(cos"e"^x)/(sin "e"^x) * "e"^x "d"x` = dt
∴ `"e"^x * cot "e"^x "d"x` = dt
∴ I = `int "t"* "dt" = "t"^2/2 + "c"`
∴ I = `([log(sin "e"^x)]^2)/2 + "c"`
APPEARS IN
RELATED QUESTIONS
Integrate : sec3 x w. r. t. x.
Integrate the function in x sin x.
Integrate the function in x (log x)2.
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following: `int logx/x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int x^3 e^(x^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate:
∫ (log x)2 dx
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
Solve: `int sqrt(4x^2 + 5)dx`
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
`intsqrt(1+x) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate:
`int1/(x^2 + 25)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate `int(1 + x + x^2/(2!))dx`.
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
Under the LIATE rule, which type of function has priority \(L\)?
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
