Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Advertisements
Solution
Let I = `int(7 + 4x +5x^2)/(2x + 3)^(3/2).dx`
= `int(5x^2 + 4x + 7)/(2x + 3)^(3/2).dx`
Put 2x + 3 = t
∴ 2dx = dt
∴ dx = `dt/(2)`
Also, x = `(t - 3)/(2)`
∴ I = `int(5((t - 3)/2)^2 + 4((t - 3)/2) + 7)/t^(3/2).dt/(2)`
= `(1)/(2) int(5((t^2 - 6t + 9)/4) + 2(t - 3) + 7)/t^(3/2)dt`
= `(1)/(2)int (5t^2 - 30t + 45 + 8t - 24 + 28)/(4t^(3/2))dt`
= `(1)/(8)int(5t^2 - 22t + 49)/t^(3/2)dt`
= `(1)/(8)int(5t^(1/2) - 22t^(-1/2) + 49t^(-3/2))dt`
= `(5)/(8)intt^(1/2)dt - 22/8 int t^(-1/2)dt + 49/8 int t^(-3/2)dt`
= `(5)/(8).t^(3/2)/((3/2)) - (11)/(4).t^(1/2)/((1/2)) + (49)/(8).t^(-1/2)/((-1/2)) + c`
= `(5)/(12)(2x+ 3)^(3/2) - (11)/(2)sqrt(2x + 3) - (49)/(4).(1)/sqrt(2x + 3) + c`.
APPEARS IN
RELATED QUESTIONS
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : tan5x
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
`int logx/x "d"x`
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
`int cos^3x dx` = ______.
Write `int cotx 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)`.
`int secx/(secx - tanx)dx` equals ______.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int sin^2(x/2)dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int(1+x+x^2/(2!))dx`
