Advertisements
Advertisements
Question
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`.
Options
True
False
Advertisements
Solution
False
Explanation:
Let I = `int "x" * "e"^"2x"` dx
`= "x" int "e"^"2x" * "dx" - int ["d"/"dx" ("x") int "e"^"2x" * "dx"]` dx
`= "x" * "e"^"2x"/2 - int 1 * "e"^"2x"/2 * "dx"`
`= "x"/2 "e"^"2x" - 1/2 int "e"^"2x" +` c
`= "x"/2 "e"^"2x" - 1/2 * "e"^"2x"/2` + c
`= "e"^"2x" ("x"/2 - 1/4)` + c
`= "e"^"2x" (("2x" - 1)/4)` + c
∴ f(x) = `(2"x" - 1)/4`
APPEARS IN
RELATED QUESTIONS
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Evaluate the following:
`int sinx/(sin 3x) dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate:
`int sin^2(x/2)dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate `int 1/(x(x-1)) dx`
