Advertisements
Advertisements
प्रश्न
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Advertisements
उत्तर
Let I = `int cos 3x cos 2x cos x *dx`
Consider cos 3x cos 2x cos x = `(1)/(2) cos 3x [2 cos 2x cos x]`
= `(1)/(2)cos3x [cos(2x + x) + cos(2x - x)]`
= `(1)/(2)[cos^2 3x + cos3x cosx]`
= `(1)/(4)[2cos^2 3x + 2cos 3x cosx]`
= `(1)/(4)[1 + cos6x + cos(3x + x) + cos(3x - x)]`
= `(1)/(4)[1 + cos6x + cos4x + cos2x]`
∴ I = `(1)/(4) int[1 + cos6x + cos4x + cos2x]*dx`
= `(1)/(4) int 1*dx + 1/4 int cos6x*dx + 1/4 int cos4x*dx + 1/4 int cos2x*dx`
= `x/(4) + (1)/(4)((sin6x)/6) + 1/4((sin4x)/4) + 1/4((sin2x)/2) + c`
= `(1)/(48)[12x + 2sin 6x + 3sin 4x + 6sin2x] + c`.
APPEARS IN
संबंधित प्रश्न
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
`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 log 2x.
Integrate the function in x cos-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^2.log x.dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
`int ("d"x)/(x - x^2)` = ______
`int logx/(1 + logx)^2 "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int 1/sqrt(x^2 - a^2)dx` = ______.
`int(logx)^2dx` equals ______.
`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^2) (x^5 + 2x^3)dx`.
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Integration by parts is a method of integration based on which rule of differentiation?
Which pair is listed as Exponential in the LIATE rule?
Evaluate \[\int e^x\sin x\,dx.\]
