Advertisements
Advertisements
Question
`int sqrt(4^x(4^x + 4)) "d"x`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -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))`
`int (xdx)/((x - 1)(x - 2))` equals:
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 : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate:
`int x/((x - 1)^2(x + 2)) dx`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int sec^3x "d"x`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
`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
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
`int 1/(4x^2 - 20x + 17) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate: `int (dx)/(2 + cos x - sin x)`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
