Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Advertisements
Solution
Let I = `int (1)/(sinx.cosx + 2cos^2x).dx`
Dividing numerator and denominator of cos2x, we get
I = `int ((1/cos^2x))/(sinx/cosx + 2).dx`
= `int sec^2x/(tan x + 2).dx`
Put tan x = t
∴ sec2x dx = dt
∴ I = `int (1)/(tan + 2)dt`
= log |t + 2| + C
= log |tan x + 2| + C
APPEARS IN
RELATED QUESTIONS
Evaluate : `∫1/(3+2sinx+cosx)dx`
Solve:
dy/dx = cos(x + y)
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : sin5x.cos8x
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
`int (sin4x)/(cos 2x) "d"x`
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int sin^-1 x`dx = ?
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int(5x + 2)/(3x - 4) dx` = ______
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate the following
`int1/(x^2 +4x-5)dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
What is integration by substitution?
For \[\int\frac{\sin x}{\sin(x+a)}\,dx\], which substitution gives \[dx=dt\]?
