Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Advertisements
उत्तर
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
संबंधित प्रश्न
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Write a value of
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of
The value of \[\int\frac{1}{x + x \log x} dx\] is
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following function w.r.t. x:
x9.sec2(x10)
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`int cos sqrtx` dx = _____________
`int sqrt(1 + sin2x) dx`
`int logx/x "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int sin^-1 x`dx = ?
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
`int cos^3x dx` = ______.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int1/(x(x-1))dx`
Evaluate `int 1/(x(x-1)) dx`
