Advertisements
Advertisements
Question
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Advertisements
Solution
Let I = `int (3x + 4)/(x^2 + 6x + 5).dx`
Let 3x + 4 = `"A"[d/dx(x^2 + 6x + 5)] + "B"`
= A(2x + 6) + B
∴ 3x + 4 = 2Ax + (6A + B)
Comparing the coefficient of x and constant on both sides, we get
2A = 3 and 6A + B = 4
∴ `"A" = (3)/(2) and 6(3/2) + "B"` = 4
∴ B = – 5
∴ 3x + 4 = `(3)/(2)(2x + 6) - 5`
∴ I = `int (3/2(2x + 6) - 5)/(x^2 + 6x + 5).dx`
= `(3)/(2) int (2x + 6)/(x^2 + 6x + 5).dx - 5 int (1)/(x^2 + 6x + 5).dx`
= `(3)/(2)"I"_1 - 5"I"_2`
I1 is of the type `int (f'(x))/f(x).dx = log|f(x)| + c`
∴ `"I"_1 = log|x^2 + 6x + 5| + c_1`
I2 = `int (1)/(x^2 + 6x + 5).dx`
= `int (1)/((x^2 + 6x + 9) - 4).dx`
= `int (1)/((x + 3)^2 - 2^2).dx`
= `(1)/(2 xx 2)log|(x + 3 - 2)/(x + 3 + 2)| + c_2`
= `(1)/(4)log|(x + 1)/(x + 5)| + c_2`
∴ I = `(3)/(2)log|x^2 + 6x+ 5| - (5)/(4)log|(x + 1)/(x + 5)| + c`, where c = c + c2.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Write a value of
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/(4x^2 - 3).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
`int logx/(log ex)^2*dx` = ______.
Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int(log(logx))/x "d"x`
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
`int dx/(1 + e^-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)
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate `int 1/("x"("x" - 1)) "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)
Evaluate `int 1/(x(x-1))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).
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
