Advertisements
Advertisements
Question
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Advertisements
Solution
Let I = `int (3x + 1)/sqrt(-2x^2 + x + 3).dx`
Let 3x + 1 = `"A"[d/dx(-2x^2 + x + 3)] + "B"`
= A(2 – 2x) + B
∴ 3x + 1 = 2Ax + (2A + B)
Comparing the coefficient of x and constant on both the sides, we get
– 2A = 7 and 2A + B = 3
∴ A = `(-7)/(2) and 2(-7/2) + "B" ` = 3
∴ B = 10
∴ 7x + 3 = `(-7)/(2)(2 - 2x) + 10`
∴ I = `int ((-7)/(2)(2 - 2x) + 10)/sqrt(3 + 2x - x^2).dx`
= `(-7)/(2) int ((2 - 2x))/sqrt(3 + 2x - x^2).dx + 10 int(1)/sqrt(3 + 2x - x^2)x`
= `(-7)/(2)"I"_1 + 10"I"_2`
In I1, put 3 + 2x – x2 = t
∴ (2 – 2x)dx = dt
∴ I1 = `int (1)/sqrt(t)dt`
= `int t^(-1/2) dt`
= `t^(1/2)/(1/2) + c_1`
= `2sqrt(3 + 2x - x^2) + c_1`
I2 = `int (1)/sqrt(3 - (x^2 - 2x + 1) + 1).dx`
= `int (1)/sqrt((2)^2 - (x - 1)^2).dx`
= `sin^-1((x - 1)/2) + c_2`
`-(3)/(2) sqrt(-2x^2 + x + 3) + (7)/(4sqrt(2)) sin^-1((4x - 1)/5) + c`.
APPEARS IN
RELATED QUESTIONS
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`sin x/(1+ cos x)`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Write a value of\[\int \log_e x\ dx\].
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int x^2/sqrt(1 - x^6)` dx = ________________
`int (cos2x)/(sin^2x) "d"x`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int sec^6 x tan x "d"x` = ______.
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
`int (logx)^2/x dx` = ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate `int1/(x(x - 1))dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x^3/sqrt(1+x^4) dx`
Evaluate `int1/(x(x-1))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).
