Advertisements
Advertisements
प्रश्न
Evaluate the following : `int x.cos^3x.dx`
Advertisements
उत्तर
cos 3x = 4 cos3x – 3cos x
∴ cos 3x + 3 cos x = 4 cos3x
∴ `int cos^3x = (1)/(4) cos3x + (3)/(4) cosx`
∴ `int cos^3x.dx = (1)/(4) int cos3x.dx + (3)/(4) int cos x.dx`
= `(1)/(4)((sin3x)/3) + (3)/(4) sinx`
= `(sin3x)/(12) + (3sinx)/(4)` ...(1)
Let I = `int x cos^3x.dx`
= `x int cos^3x.dx - int[{d/dx (x) int cos^3x.dx}].dx`
= `x[(sin3x)/(12) + (3sinx)/(4)]- int 1.((sin3x)/(12) + (3sinx)/4).dx` ...[By (1)]
= `(xsin3x)/(12) + (3x sinx)/(4) - (1)/(12) int sin 3x.dx - 3/4 int sin x.dx`
= `(x sin3x)/(12) + (3xsinx)/(4) - (1)/(12) ((-cos3x)/3) - (3)/(4) (- cos x) + c`
= `(1)/(4)[x/3 sin 3x + 1/9 cos3x + 3x sin x + 3 cos x] + c`.
APPEARS IN
संबंधित प्रश्न
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
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
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in x sin x.
Integrate the function in x sin 3x.
Integrate the function in `x^2e^x`.
Integrate the function in x2 log x.
Evaluate the following : `int x.sin^2x.dx`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*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` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
`int (cos2x)/(sin^2x cos^2x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int logx/(1 + logx)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
Solve: `int sqrt(4x^2 + 5)dx`
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`int e^(ax)*cos(bx + c)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 the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
