Advertisements
Advertisements
Question
Evaluate `int x^2"e"^(4x) "d"x`
Advertisements
Solution
Let I = `int x^2*"e"^(4x) "d"x`
= `x^2 int "e"^(4x) "d"x - int ["d"/("d"x)(x^2) int"e"^(4x)"d"x]"d"x`
= `x^2* ("e"^(4x))/4 - int 2x* ("e"^(4x))/4 "d"x`
= `(x^2*"e"^(4x))/4 - 1/2 intx*"e"^(4x)"d"x`
= `(x^2"e"^(4x))/4 - 1/2[x"f""e"^(4x)"d"x - int("d"/("d"x)(x) int"e"^(4x)"d"x)"d"x]`
= `(x^2"e"^(4x))/4 - 1/2[x* ("e"^(4x))/4 - int 1* ("e"^(4x))/4 "d"x]`
= `(x^2"e"^(4x))/4 - 1/2[(x*"e"^(4x))/4 - 1/4 int"e"^(4x)"d"x]`
= `(x^2"e"^(4x))/4 - 1/2[(x"e"^(4x))/4 - 1/4*("e"^(4x))/4] + "c"`
= `(x^2"e"^(4x))/4 - (x"e"^(4x))/8 + ("e"^(4x))/32 + "c"`
∴ I = `("e"^(4x))/4[x^2 - x/2 + 1/8] + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
`int x^2sqrt("a"^2 - x^6) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
`int 1/(4x^2 - 20x + 17) "d"x`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
