Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
उत्तर
Let I = `int (2x + 3)/(2x^2 + 3x - 1).dx`
Let 2x + 3 = `"A"[d/dx(2x^2 + 3x - 1)] + "B"`
= A(4x + 3) + B
∴ 2x + 3 = 4Ax + (3A + B)
Comapring the coefficientof x and constant on both sides, we get
4A = 2 and 3A + B = 3
∴ A = `(1)/(2) and 3(1/2) + "B"` = 3
∴ B = `(3)/(2)`
∴ 2x + 3 = `(1)/(2)(4x + 3) + (3)/(2)`
∴ I = `int (1/2(4x + 3) + (3)/(2))/(2x^2 + 3x - 1).dx`
= `(1)/(2) int (4x + 3)/(2x^2 + 3x - 1).dx + (3)/(2) int (1)/(2x^2 + 3x - 1).dx`
= `(1)/(2)"I"_1 + (3)/(2)"I"_2`
I1 is of the type `int (f'(x))/f(x)dx = log|f(x)| + c`
∴ I1 = log |2x2 + 3x – 1| + c1
I2 = `int (1)/(2x^2 + 3x - 1).dx`
= `(1)/(2) int (1)/(x^2 + 3/2x - 1/2).dx`
= `(1)/(2) int (1)/((x^2 + 3/2x + 9/16) - 9/16 - 1/2).dx`
= `(1)/(2) int (1)/((x + 3/4)^2 - (sqrt(17)/4)^2).dx`
= `(1)/(2) xx (1)/(2 xx sqrt(17)/(4))log|(x + 3/4 - sqrt(17)/4)/(x + 3/4 + sqrt(17)/4)| + c_2`
= `(1)/sqrt(17)log|(4x + 3 - sqrt(17))/(4x + 3 + sqrt(17))| + c_2`
∴ I = `(1)/(2)log|2x^2 + 3x - 1| + (3)/(2sqrt(17))log|(4x + 3 - sqrt(17))/(4x + 3 + sqrt(17))| + c`, where c = c + c2.
APPEARS IN
संबंधित प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Evaluate :`intxlogxdx`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`sqrt(tanx)/(sinxcos x)`
Integrate the functions:
`(1+ log x)^2/x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
`int logx/(log ex)^2*dx` = ______.
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
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 (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
`int sqrt(1 + "x"^2) "dx"` =
Evaluate: `int "e"^sqrt"x"` dx
`int cos sqrtx` dx = _____________
`int (cos2x)/(sin^2x) "d"x`
`int cot^2x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int (cos x)/(1 - sin x) "dx" =` ______.
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
If f'(x) = `x + 1/x`, then f(x) is ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
`int (logx)^2/x dx` = ______.
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate `int(1+ x + x^2/(2!)) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
`int "cosec"^4x dx` = ______.
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int(1+x+x^2/(2!))dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
