Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int cos^2x.dx`
उत्तर
Recall the identity cos 2x = 2 cos2x – 1, which gives
`cos^2x = (1 + cos2x)/(2)`
Therefore, `int cos^2 x.dx`
= `(1)/(2)int (1 + cos 2x).dx`
= `(1)/(2)int dx + (1)/(2) int cos 2x .dx`
= `x/(2) + (1)/(4)sin 2x + C`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Find : `int(x+3)sqrt(3-4x-x^2dx)`
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
The value of \[\int\frac{1}{x + x \log x} dx\] is
Evaluate the following integrals : tan2x dx
Evaluate the following integrals : `int(4x + 3)/(2x + 1).dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
`int logx/(log ex)^2*dx` = ______.
Integrate the following w.r.t.x : `(3x + 1)/sqrt(-2x^2 + x + 3)`
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
`int sqrt(1 + "x"^2) "dx"` =
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, 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 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int 1/(sqrt("x") + "x")` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int sqrt(x^2 + 2x + 5)` dx = ______________
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int (cos x)/(1 - sin x) "dx" =` ______.
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
`int cos^3x dx` = ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate `int1/(x(x - 1))dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int sin^2(x/2)dx`
Evaluate:
`int(cos 2x)/sinx dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int1/(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 1/(x(x-1)) dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
