Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int cos^2x.dx`
Advertisements
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
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`sin x/(1+ cos x)`
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
`int 1/(cos x - sin x)` dx = _______________
`int x^2/sqrt(1 - x^6)` dx = ________________
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate `int(1+ x + x^2/(2!)) dx`
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate `int(1 + x + x^2/(2!))dx`
`int x^3 e^(x^2) dx`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int 1/(x(x-1))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`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
