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
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
cot x log sin x
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
