Advertisements
Advertisements
Question
Evaluate the following : `int x.cos^3x.dx`
Advertisements
Solution
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
RELATED QUESTIONS
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`
Integrate the function in x log 2x.
Integrate the function in x (log x)2.
Integrate the function in ex (sinx + cosx).
Integrate the function in `(xe^x)/(1+x)^2`.
`intx^2 e^(x^3) dx` equals:
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 the following : `int x.sin^2x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
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 (sin^m x)/(cos^(m+2)x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : e2x sin x cos x
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 (log "x")/(1 + log "x")^2` 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)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int logx/(1 + logx)^2 "d"x`
∫ log x · (log x + 2) dx = ?
`int log x * [log ("e"x)]^-2` dx = ?
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
`int tan^-1 sqrt(x) "d"x` is equal to ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Solve: `int sqrt(4x^2 + 5)dx`
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`intsqrt(1+x) dx` = ______
Solve the following
`int_0^1 e^(x^2) x^3 dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate `int tan^-1x dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate:
`int x^2 cos x dx`
