Advertisements
Advertisements
Question
Evaluate the following : `int cos sqrt(x).dx`
Advertisements
Solution
Let I = `int cos sqrt(x).dx`
Put `sqrt(x) = t`
∴ x = t2
∴ dx = 2t .dt
∴ I = `int(cost)2t.dt`
= `int 2t cos t.dt`
= `2t int cos.dt - int [d/dt (2t) int cos t.dt ].dt`
= `2tsint - int 2 sint.dt`
= 2t sin t + 2 cos t + c
= `2[sqrt(x)sinsqrt(x) + cos sqrt(x)] + c`.
APPEARS IN
RELATED QUESTIONS
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate : sec3 x w. r. t. x.
Integrate the function in x tan-1 x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
Evaluate:
∫ (log x)2 dx
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int 1/sqrt(x^2 - 9) dx` = ______.
`int(logx)^2dx` equals ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Solution of the equation `xdy/dx=y log y` is ______
`inte^(xloga).e^x dx` is ______
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate:
`inte^x sinx dx`
Evaluate:
`int (logx)^2 dx`
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 `int tan^-1x dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
