Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Solution
Let I = `int (x + 1)sqrt(2x^2 + 3)`
Let x + 1 = `"A"[d/dx (2x^2 + 3)] + "B"`
= A (4x) + B
= 4Ax + B
Comparing the coefficients of and constant on both sides, we get
4A = 1, B = 1
∴ A = `(1)/(4), "B"` = 1
∴ x + 1 = `(1)/(4)(4x) + 1`
∴ I = `int [1/4 (4x) + 1]sqrt(2x^2 + 3).dx`
= `(1)/(4) int 4x sqrt(2x^2 + 3).dx + int sqrt(2x^2 + 3).dx`.
= I1 + I2
In I1 = put 2x2 + 3 = t
∴ 4x.dx = dt
∴ I1 = `(1)/(4) int t^(12).dt`
= `(1)/(4)(t^(3/2)/(3/2)) + c_1`
= `(1)/(6)(2x^2 + 3)^(3/2) + c_1`
I2 = `int sqrt(2x^2 + 3).dx`
= `sqrt(2) int sqrt(x^2 + 3/2).dx`
= `sqrt(2)[x/2sqrt(x^2 + 3/2) + ((3/2))/(2)log|x + sqrt(x^2 + 3/2)|] + c_2`
= `sqrt(2)[x/2sqrt(x^2 + 3/2) + (3)/(4)log|x + sqrt(x^2 + 3/2)|] + c_2`
∴ I = `(1)/(6)(2x^2 + 3)^(3/2) + sqrt(2)[x/2 sqrt(x^2 + 3/2) + (3)/(4) log|x + sqrt(x^2 + 3/2)|] + c`, where c = c1 + c2.
APPEARS IN
RELATED QUESTIONS
Prove that:
`int sqrt(a^2 - x^2) dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a)+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
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in `x^2e^x`.
Integrate the function in x2 log x.
Integrate the function in x sec2 x.
Integrate the function in tan-1 x.
Integrate the function in x (log x)2.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
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/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)]`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
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`
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
`int logx/(1 + logx)^2 "d"x`
∫ log x · (log x + 2) dx = ?
`int log x * [log ("e"x)]^-2` dx = ?
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x 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`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
The value of `inta^x.e^x dx` equals
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
