Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Advertisements
उत्तर
Let I = `int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Let `(3"e")^"2t" + 5 = "A"(4"e"^"2t" - 5) + "B" "d"/"dt" (4"e"^"2t" - 5)`
`= 4 "Ae"^"2t" - 5"A" + "B"(8"e"^"2t")`
∴ `(3"e")^"2t" + 5 = (4"A" + 8"B") "e"^"2t" - 5"A"`
Comparing the coefficients of e2t and constant term on both sides, we get
4A + 8B = 3 and - 5A = 5
Solving these equations, we get
A = - 1 and B = `7/8`
∴ I = `int (- 1(4"e"^"2t" - 5) + 7/8 (8"e"^"2t"))/(4"e"^"2t" - 5)` dt
`= - int "dt" + 7/8 int (8"e"^"2t")/(4"e"^"2t" - 5)` dt
∴ I = `- "t" + 7/8 log |4"e"^"2t" - 5|` + c .....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`x/(e^(x^2))`
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Evaluate: `int "e"^sqrt"x"` dx
`int 1/(cos x - sin x)` dx = _______________
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int (sin4x)/(cos 2x) "d"x`
`int(log(logx))/x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int ("d"x)/(x(x^4 + 1))` = ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int 1/(sinx.cos^2x)dx` = ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate.
`int (5x^2 -6x + 3)/(2x -3)dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
Which substitution is appropriate for \[\sqrt{\frac{x}{x-a}}\], \[\sqrt{\frac{x-a}{x}}\], \[\sqrt{x(x-\mathrm{a})}\], or \[\frac{1}{\sqrt{x(x-\mathrm{a})}}\]?
After choosing \[u=g(x)\], what is the next step?
How should a substitution be chosen?
