Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Advertisements
उत्तर
Let I = `int x.sin 2x. cos 5x.dx`
`sin 2x cos 5x = (1)/(2)[2 sin2x cos5x]`
= `(1)/(2)[sin(2x+ 5x) + sin(2x - 5x)]`
= `(1)/(2)[sin7x - sin3x]`
∴ `int sin 2x cos 5x .dx = (1)/(2)[int sin 7x ..dx - intsin 3x.dx]`
= `(1)/(2)((-cos7x)/7) - (1)/(2) ((- cos3x)/3)`
= `-(1)/(14) cos7x + (1)/(6) cos3x` ...(1)
I = `int x sin 2x cos 5x.dx`
= `x int sin 2x cos 5x.dx - int [d/dx (x) int sin 2x cos 5x.dx].dx`
= `x[-1/14 cos7x + 1/6 cos 3x] - int 1.(-1/14 cos7x + 1/6 cos3x).dx` ...[By (1)]
= `-x/(14) cos7x + x/(6) cos3x + (1)/(14) int cos7x.dx - (1)/(6) int cos 3x.dx`
= `-x/(14) cos7x + x/(6) cos3x + (1)/(14) ((sin7x)/7) - (1)/(6) ((sin3x)/3) + c`
= `- x/(14) cos7x + x/(6) cos3x + (sin7x)/(98) - (sin3x)/(18) + c`.
APPEARS IN
संबंधित प्रश्न
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
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in `x^2e^x`.
Integrate the function in x tan-1 x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
`intx^2 e^(x^3) dx` equals:
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 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)/(x^3 sqrt(x^2 - 1)`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
`int sin4x cos3x "d"x`
Evaluate `int 1/(x(x - 1)) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) 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.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int(logx)^2dx` equals ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
`intsqrt(1+x) dx` = ______
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate:
`int e^(logcosx)dx`
Evaluate:
`int (logx)^2 dx`
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
`∫ sin^(−1)` xdx is equal to ______.
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
Repeated parts may be needed for which pair of integrals?
