Advertisements
Advertisements
प्रश्न
`int sqrt(4^x(4^x + 4)) "d"x`
Advertisements
उत्तर
Let I = `int sqrt(4^x(4^x + 4)) "d"x`
= `int sqrt((2^x)^2 [(2^x)^2 + 4]) "d"x`
= `int sqrt((2^x)^2 + 2^2)*2^x "d"x`
Put 2x = t
∴ 2x log2 dx = dt
∴ 2x dx = `1/(log 2) "dt"`
∴ I = `1/(log 2) int sqrt("t"^2 + 2^2) "dt"`
= `1/(log 2)["t"/2 sqrt("t"^2 + 2^2) + 2^2/2log |"t" + sqrt("t"^2 + 2^2)|] + "c"`
∴ I = `1/(log 2) [2^x/2 sqrt(4x + 4) + 2log |2^x + sqrt(4^x + 4)|] + "c"`
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int x^2"e"^(4x) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
