Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Advertisements
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
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
`int (dx)/(sin^2 x cos^2 x)` equals:
Write a value of
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following : `int (logx)2.dx`
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
`int (log x)/(log ex)^2` dx = _________
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int cos^7 x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int(1 - x)^(-2) dx` = ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate the following.
`int "x"^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.
`int1/(x^2 + 4x-5)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
For \[\sqrt{\frac{x-\alpha}{\beta-x}}\] or \[\sqrt{(x-\alpha)(\beta-x)}\], where \[\beta>\alpha\], which substitution is used?
In \[\int\sin^3x\cos^2x\,dx\], which rewriting prepares the integrand for the substitution \[t=\cos x\]?
