Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(sinx cos^3x)/(1 + cos^2x)`
उत्तर
Let I = `int (sinx cos^3x)/(1 + cos^2x).dx`
Put cos x = t
∴ – sin x dx = dt
∴ sin x dx = – dt
I = `- int t^3/(t^2 + 1)dt`
= `- int (t(t^2 + 1) - t)/(t^2 + 1)dt`
= `- int[(t(t^2 + 1))/(t^2 + 1) - t/(t^2 + 1)]dt`
= `- int t dt + int t/(t^2 + 1)dt`
= `- int t dt + (1)/(2) int (2t)/(t^2 + 1)dt`
= `t^2/(2) + (1)/(2)log|t^2 + 1| + c`
... `[∵ d/dt(t^2 + 1) = 2t and int (f'(x))/f(x)dx = log [f(x)] + c]`
= `-(1)/(2) cos^2x + (1)/(2)log|cos^2x + 1| + c`
= `(1)/(2)[log|cos^2x + 1| - cos^2x] + c`.
APPEARS IN
संबंधित प्रश्न
Find : `int(x+3)sqrt(3-4x-x^2dx)`
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`1/(1 + cot x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
`int sqrt(1 + "x"^2) "dx"` =
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Evaluate: `int log ("x"^2 + "x")` dx
`int sqrt(1 + sin2x) "d"x`
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
`int cos^3x dx` = ______.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
if `f(x) = 4x^3 - 3x^2 + 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(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate `int1/(x(x - 1))dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
`int x^3 e^(x^2) dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
Evaluate `int 1/(x(x-1)) dx`
