Advertisements
Advertisements
Question
`int e^x sec x (1 + tan x) dx` equals:
Options
ex cos x + C
ex sec x + C
ex sin x + C
ex tan x + C
Advertisements
Solution
ex sec x + C
Explanation:
Let `I = int e^x sec x (1 + tan x) dx`
`= int e^x (sec x + sec x tan x) dx`
` = int (sec x) e^x dx + int e^x sec x tan x dx`
`= I_1 + int e^x sec x tan x` .... (1)
`I_1 = int (sec x)e^x dx`
`I_1 = (sec x) int e^x dx - int (sec x tan x int e^x dx) dx`
`= (sec x) e^x - int e^x sec x tan x dx`
Putting this value in equation (1),
`I = I_1 + int e^x sec x tan x dx`
`= (sec x) e^x - int e^x sec x tan x dx + int e^x sec x tan x dx + C`
`= e^x sec x + C`
APPEARS IN
RELATED QUESTIONS
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x (log x)2.
Integrate the function in (x2 + 1) log x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int sin4x cos3x "d"x`
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`intsqrt(1+x) dx` = ______
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
`int1/(x+sqrt(x)) dx` = ______
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate `int tan^-1x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
The value of `inta^x.e^x dx` equals
`∫ sin^(−1)` xdx is equal to ______.
