Advertisements
Advertisements
Question
Evaluate the following:
`int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`
Solution
Let I = `int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`
Put `tan^-1x` = t
⇒ `1/(1 + x^2) * "d"x` = dt
= `int "e"^"t" (1 + tan "t" + tan^2 "t")"dt"`
= `int "e"^"t" (sec^2 "t" + tan "t")"dt"`
Here f(t) = tan t
∴ f'(t) = sec2t
= `"e"^"t" * "f"("t")`
= `"e"^"t" tan "t"`
= `"e"^(tan^-1x) * x + "c"` ....`[because int "e"^x ["f"(x) + "f'"(x)]"d"x = "e"^2"f"(x) + "C"]`
Hence, I = `"e"^(tan^-1x) * x + "C"`.
APPEARS IN
RELATED QUESTIONS
Find the integrals of the function:
sin2 (2x + 5)
Find the integrals of the function:
sin3 (2x + 1)
Find the integrals of the function:
sin 4x sin 8x
Find the integrals of the function:
`(1-cosx)/(1 + cos x)`
Find the integrals of the function:
`(cos 2x - cos 2 alpha)/(cos x - cos alpha)`
Find the integrals of the function:
`(cos x - sinx)/(1+sin 2x)`
Find the integrals of the function:
tan3 2x sec 2x
Find the integrals of the function:
tan4x
Find the integrals of the function:
`(sin^3 x + cos^3 x)/(sin^2x cos^2 x)`
Find the integrals of the function:
`(cos 2x+ 2sin^2x)/(cos^2 x)`
Find the integrals of the function:
`1/(sin xcos^3 x)`
Differentiate : \[\tan^{- 1} \left( \frac{1 + \cos x}{\sin x} \right)\] with respect to x .
Evaluate : \[\int\limits_0^\pi \frac{x \tan x}{\sec x \cdot cosec x}dx\] .
Find `int_ (log "x")^2 d"x"`
Find `int_ (sin2"x")/((sin^2 "x"+1)(sin^2"x"+3))d"x"`
Find: `int_ (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`
Find: `int sin^-1 (2x) dx.`
Evaluate `int tan^8 x sec^4 x"d"x`
`int "dx"/(sin^2x cos^2x)` is equal to ______.
Evaluate the following:
`int ((1 + cosx))/(x + sinx) "d"x`
Evaluate the following:
`int (cosx - cos2x)/(1 - cosx) "d"x`
Evaluate the following:
`int sin^-1 sqrt(x/("a" + x)) "d"x` (Hint: Put x = a tan2θ)
The value of the integral `int_(1/3)^1 (x - x^3)^(1/3)/x^4 dx` is
