Advertisements
Advertisements
प्रश्न
Evaluate the following : `int cos sqrt(x).dx`
Advertisements
उत्तर
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
संबंधित प्रश्न
Integrate the function in x sin−1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in x sec2 x.
Integrate the function in tan-1 x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int x^3 e^(x^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
Evaluate:
∫ (log x)2 dx
`int 1/(4x + 5x^(-11)) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/x "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
Find `int_0^1 x(tan^-1x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int1/(x+sqrt(x)) dx` = ______
Evaluate `int tan^-1x dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
