Advertisements
Advertisements
Question
Integrate the function:
`(2+ sin 2x)/(1+ cos 2x) e^x`
Solution
Let `I = int (2 + sin 2x)/(1 + cos 2x) e^x dx`
`int (2 + 2 sin x cos x)/(1 + 2 cos^2 x - 1)e^x dx`
`= int (2 (1 + sin x cos x))/(2 cos^2 x) e^x dx`
`= int (sec^2 x * e^x + tan * e^x) dx`
`= int e^x (sec^2 x + tan x) dx`
Putting ex tan x = t
(ex sec2 x + tan x · ex)dx = dt
Hence, I = `int 1 * dt`
= t + C = ex tan x + C
APPEARS IN
RELATED QUESTIONS
Find an anti derivative (or integral) of the following function by the method of inspection.
sin 2x
Find the following integrals:
`int (4e^(3x) + 1)`
Find the following integrals:
`intx^2 (1 - 1/x^2)dx`
Find the following integrals:
`int (ax^2 + bx + c) dx`
Find the following integrals:
`int(sqrtx - 1/sqrtx)^2 dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Find the following integrals:
`int(1 - x) 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:
`int(2x^2 - 3sinx + 5sqrtx) dx`
Find the following integrals:
`int (2 - 3 sinx)/(cos^2 x) dx.`
The anti derivative of `(sqrtx + 1/ sqrtx)` equals:
If `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0, then f(x) is ______.
Integrate the function:
`1/(x - x^3)`
Integrate the function:
`1/(x^2(x^4 + 1)^(3/4))`
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:
`1/((x^2 + 1)(x^2 + 4))`
Integrate the function:
f' (ax + b) [f (ax + b)]n
The anti derivative of `(sqrt(x) + 1/sqrt(x))` is equals:
`int (e^x (1 + x))/(cos^2 (xe^x)) dx` equal
`int (dx)/sqrt(9x - 4x^2)` equals
`int (xdx)/((x - 1)(x - 2))` equals
`f x^2 e^(x^3) dx` equals
`int e^x sec x(1 + tanx) dx` equals
`int sqrt(1 + x^2) dx` is equal to
`d/(dx)x^(logx)` = ______.
`int (dx)/sqrt(5x - 6 - x^2)` equals ______.
Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.
