Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Advertisements
उत्तर
Let I = `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
= `int sqrt((e^(2x)(e^x - 1))/(e^x + 1)).dx`
= `int e^xsqrt((e^x - 1)/(e^x + 1)).dx`
Put ex = t
∴ ex dx = dt
∴ I = `int sqrt((t - 1)/(t + 1))dt`
= `int sqrt((t - 1)/(t + 1) xx (t - 1)/(t - 1))dt`
= `int sqrt(((t - 1)^2)/(t^2 - 1)dt`
= `int (t - 1)/sqrt(t^2 - 1)dt`
= `(1)/(2) int (2t)/sqrt(t^2 - 1)dt - int (1)/sqrt(t^2 - 1)dt`
= I1 – I2
In I1, put t2 – 1 = θ
∴ 2t dt = dθ
∴ I1 = `(1)/(2)int (dθ)/sqrt(θ)`
= `(1)/(2) int θ^(-1/2) dθ`
= `(1)/(2) (θ^(1/2))/((1/2)) + c_1`
= `sqrt(θ) + c_1`
= `sqrt(t^2 - 1) + c_1`
= `sqrt(e^(2x) - 1) + c_1`
and I2 = `int (1)/sqrt(t^2 - 1)dt`
= `log|t + sqrt(t^2 - 1)| + c_2`
= `log|e^x + sqrt(e^(2x) - 1)| + c_2`
∴ I = `sqrt(e^(2x) - 1) - log|e^x + sqrt(e^(2x) - 1) + c`, where c = c1 + c2.
APPEARS IN
संबंधित प्रश्न
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`(e^(2x) - 1)/(e^(2x) + 1)`
Integrate the functions:
cot x log sin x
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Evaluate `int 1/("x" ("x" - 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
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
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 "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate: `int "e"^sqrt"x"` dx
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int x^x (1 + logx) "d"x`
`int(log(logx))/x "d"x`
`int(sin2x)/(5sin^2x+3cos^2x) dx=` ______.
`int sec^6 x tan x "d"x` = ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
`int cos^3x dx` = ______.
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
`int x^3 e^(x^2) dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate the following.
`int x^3/sqrt(1+x^4) 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).
