Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Advertisements
Solution
Let I = `int sec^2x.sqrt(tan^2x + tan x - 7)`
Put tan x = t
∴ sec2x.dx = dt
∴ I = `int sqrt(t^2 + t - 7).dt`
= `int sqrt(t^2 + t + 1/4 - 29/4).dt`
= `int sqrt((t + 1/2)^2 - (sqrt(29)/2)^2).dt`
= `((t + 1/2)/2) sqrt((t + 1/2)^2 - 29/4) - ((29/4))/(2)log|(t + 1/2) + sqrt((t + 1/2)^2 - 29/4)| + c`
= `((2t + 1))/(4)sqrt(t^2 + t - 7) - (29)/(8)log|(t + 1/2) + sqrt(t^2 + t - 7)| + c`
= `((2tanx + 1)/4)sqrt(tan^2x + tanx - 7) - (29)/(8)log|(tanx + 1/2) + sqrt(tan^2x + tanx - 7)| + 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`
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
Integrate the function in x tan-1 x.
Integrate the function in (sin-1x)2.
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
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^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
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 cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
`int e^x (1/x - 1/x^2)`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"` =
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int 1/(4x^2 - 1) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) dx` = ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate the following.
`intx^3 e^(x^2) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^2e^(4x)dx`
