Advertisements
Advertisements
Question
`int sin4x cos3x "d"x`
Solution
Let I = `int sin 4x * cos3x "d"x`
= `1/2 int (2 sin 4x * cos 3x) "d"x`
= `1/2 int [sin (4x + 3x) + sin(4x - 3x)] "d"x` .......[∵ 2 sin A cos B = sin(A + B) + sin(A − B)]
= `1/2 int (sin 7x + sin x) "d"x`
= `1/2 [int sin7 x "d"x + int sin x "d"x]`
= `1/2((-cos7x)/7 - cos x) + "c"`
∴ I = `- 1/14 cos 7x - 1/2 cos x + "c"`
APPEARS IN
RELATED QUESTIONS
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin 3x.
Integrate the function in (sin-1x)2.
Integrate the function in x sec2 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^2 sin 3x dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
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]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*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` =
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "x"^2 "e"^"3x"`dx
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int ("d"x)/(x - x^2)` = ______
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
`int tan^-1 sqrt(x) "d"x` 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: `int e^x.sin2xdx`
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Solve: `int sqrt(4x^2 + 5)dx`
`int(logx)^2dx` equals ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int(1-x)^-2 dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
`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
`inte^(xloga).e^x dx` is ______
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
`int logx dx = x(1+logx)+c`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
