Advertisements
Advertisements
Question
Integrate the following w.r.t.x : sec4x cosec2x
Advertisements
Solution
Let I = `int sec^4x "cosec"^2x*dx`
= `int sec^4x "cosec"^2x* sec^2x*dx`
Put tan x = t
∴ sec2x·dx = d
Also, sec2x cosec2x = (1 + tan2x)(1 + cot2x)
= `(1 + t^2)(1 + 1/t^2)`
= `(1 + t^2)((t^2 + 1)/t^2)`
= `(t^4 + 2t^2 + 1)/t^2`
= `t^2 + 2 + (1)/t^2`
∴ I = `int (t^2 + 2 + 1/t^2)*dt`
= `int t^2*dt + 2 int *dt + int 1/t^2*dt`
= `t^3/(3) + 2t + (t^-1)/((-1)) + c`
= `(1)/(3)tan^3x + 2tanx - (1)/tanx + c`
= `(1)/(3cot^3x) + (2)/(cotx) - cot x + c`.
APPEARS IN
RELATED QUESTIONS
Integrate : sec3 x w. r. t. x.
Integrate the function in x sin 3x.
Integrate the function in x2 log x.
Integrate the function in x (log x)2.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*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 : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : e2x sin x cos x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
∫ x log x dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int(x + 1/x)^3 dx` = ______.
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
Evaluate `int 1/(x log x) "d"x`
Evaluate `int 1/(4x^2 - 1) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
Solve: `int sqrt(4x^2 + 5)dx`
`int(logx)^2dx` equals ______.
`int_0^1 x tan^-1 x dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
`int(1-x)^-2 dx` = ______
Evaluate:
`inte^x sinx dx`
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`
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^3e^(x^2) dx`
