Advertisements
Advertisements
Question
Integrate the function:
`1/(cos (x+a) cos(x+b))`
Advertisements
Solution
Let I = `1/(cos (x + a) cos (x + b)) dx`
`= 1/(sin (a - b)) int (sin [(x + a) - (x + b)])/(cos (x + a) cos (x + b))`dx
`= 1/(sin (a + b)) int (sin (x + a) cos (x + b) - cos (x + a) sin (x + b))/(cos (x + a) cos (x + b)) dx`
`= 1/(sin (a - b)) int [(sin (x + a))/(cos (x + a)) - (sin (x + b))/(cos (x + b))] dx`
`= 1/(sin (a - b)) int [tan (x + a) - tan (x + b)] dx`
`= 1/(sin (a - b)) = [log |cos (x + a)| - log |cos (x + b)|] + C`
`= 1/(sin (a - b)) = [log |sec(x + a)| - log |sec(x + b)|] + C`
`= 1/(sin (a - b)) log |(sec(x + a))/(sec(x + b))| + C`
or `= 1/(sin (a - b)) log |(cos (x + a))/(cos (x + b))| + C`
APPEARS IN
RELATED QUESTIONS
Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`
If `f(x) =∫_0^xt sin t dt` , then write the value of f ' (x).
Find the following integrals:
`int (4e^(3x) + 1)`
Find the following integrals:
`intx^2 (1 - 1/x^2)dx`
Find the following integrals:
`int(sqrtx - 1/sqrtx)^2 dx`
Find the following integrals:
`int (x^3 + 5x^2 -4)/x^2 dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Find the following integrals:
`intsqrtx( 3x^2 + 2x + 3) dx`
Find the following integrals:
`int(2x - 3cos x + e^x) dx`
Find the following integrals:
`intsec x (sec x + tan x) dx`
Find the following integrals:
`int (2 - 3 sinx)/(cos^2 x) dx.`
If `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0, then f(x) is ______.
Integrate the function:
`1/(sqrt(x+a) + sqrt(x+b))`
Integrate the function:
`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`
Integrate the function:
`(e^(5log x) - e^(4log x))/(e^(3log x) - e^(2log x))`
Integrate the function:
`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`
Integrate the function:
`x^3/(sqrt(1-x^8)`
Integrate the function:
`e^(3log x) (x^4 + 1)^(-1)`
Evaluate `int tan^(-1) sqrtx dx`
Evaluate: `int (1 - cos x)/(cos x(1 + cos x)) dx`
If `d/(dx) f(x) = 4x^3 - 3/x^4`, such that `f(2) = 0`, then `f(x)` is
`int (sin^2x - cos^2x)/(sin^2x cos^2x) dx` is equal to
`int (dx)/sqrt(9x - 4x^2)` equal
`int (dx)/sqrt(9x - 4x^2)` equals
`int (xdx)/((x - 1)(x - 2))` equals
`int x^2 e^(x^3) dx` equals
`int e^x sec x(1 + tanx) dx` equals
What is anti derivative of `e^(2x)`
If the normal to the curve y(x) = `int_0^x(2t^2 - 15t + 10)dt` at a point (a, b) is parallel to the line x + 3y = –5, a > 1, then the value of |a + 6b| is equal to ______.
`d/(dx)x^(logx)` = ______.
If y = `x^((sinx)^(x^((sinx)^(x^(...∞)`, then `(dy)/(dx)` at x = `π/2` is equal to ______.
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
