Problem to solve
Find the indefinite integral \[\int\frac{\sqrt{[3x^2+4]^3}}{x^4} dx\]
Solution
STEP 1
Take the integral: integral(3 x^2 + 4)^(3/2)/x^4 dx
STEP 2
For the integrand (3 x^2 + 4)^(3/2)/x^4, substitute x = (2 tan(u))/sqrt(3) and dx = (2 sec^2(u))/sqrt(3) du. Then (3 x^2 + 4)^(3/2) = (4 tan^2(u) + 4)^(3/2) = 8 sec^3(u) and u = tan^(-1)((sqrt(3) x)/2): = 2/sqrt(3) integral9/2 csc^4(u) sec(u) du
STEP 3
Factor out constants: = 3 sqrt(3) integral csc^4(u) sec(u) du
STEP 4
For the integrand csc^4(u) sec(u), use the trigonometric identity csc^2(u) = cot^2(u) + 1: = 3 sqrt(3) integral(cot^2(u) + 1)^2 sec(u) du
STEP 5
Expanding the integrand (cot^2(u) + 1)^2 sec(u) gives sec(u) + cot^3(u) csc(u) + 2 cot(u) csc(u): = 3 sqrt(3) integral(sec(u) + cot^3(u) csc(u) + 2 cot(u) csc(u)) du
STEP 6
Integrate the sum term by term and factor out constants: = 3 sqrt(3) integral sec(u) du + 3 sqrt(3) integral cot^3(u) csc(u) du + 6 sqrt(3) integral cot(u) csc(u) du
STEP 7
The integral of sec(u) is log(tan(u) + sec(u)): = 3 sqrt(3) log(tan(u) + sec(u)) + 3 sqrt(3) integral cot^3(u) csc(u) du + 6 sqrt(3) integral cot(u) csc(u) du
STEP 8
For the integrand cot^3(u) csc(u), use the trigonometric identity cot^2(u) = csc^2(u) - 1: = 3 sqrt(3) log(tan(u) + sec(u)) + 3 sqrt(3) integral cot(u) csc(u) (csc^2(u) - 1) du + 6 sqrt(3) integral cot(u) csc(u) du
STEP 9
For the integrand cot(u) csc(u) (csc^2(u) - 1), substitute s = csc(u) and ds = -cot(u) csc(u) du: = 3 sqrt(3) log(tan(u) + sec(u)) + -3 sqrt(3) integral(s^2 - 1) ds + 6 sqrt(3) integral cot(u) csc(u) du
STEP 10
Integrate the sum term by term and factor out constants: = 3 sqrt(3) log(tan(u) + sec(u)) + -3 sqrt(3) integral s^2 ds + 3 sqrt(3) integral1 ds + 6 sqrt(3) integral cot(u) csc(u) du
STEP 11
The integral of s^2 is s^3/3: = -sqrt(3) s^3 + 3 sqrt(3) log(tan(u) + sec(u)) + 3 sqrt(3) integral1 ds + 6 sqrt(3) integral cot(u) csc(u) du
STEP 12
The integral of 1 is s: = 3 sqrt(3) s - sqrt(3) s^3 + 3 sqrt(3) log(tan(u) + sec(u)) + 6 sqrt(3) integral cot(u) csc(u) du
STEP 13
The integral of cot(u) csc(u) is -csc(u): = -sqrt(3) s^3 + 3 sqrt(3) s - 6 sqrt(3) csc(u) + 3 sqrt(3) log(tan(u) + sec(u)) + constant
STEP 14
Substitute back for s = csc(u): = -sqrt(3) csc^3(u) - 3 sqrt(3) csc(u) + 3 sqrt(3) log(tan(u) + sec(u)) + constant
STEP 15
Substitute back for u = tan^(-1)((sqrt(3) x)/2): = -sqrt(3) csc(tan^(-1)((sqrt(3) x)/2))^3 - 3 sqrt(3) csc(tan^(-1)((sqrt(3) x)/2)) + 3 sqrt(3) log(tan(tan^(-1)((sqrt(3) x)/2)) + sec(tan^(-1)((sqrt(3) x)/2))) + constant
STEP 16
Simplify using csc(tan^(-1)(z)) = sqrt(z^2 + 1)/z, sec(tan^(-1)(z)) = sqrt(z^2 + 1) and tan(tan^(-1)(z)) = z: = 3 sqrt(3) log(1/2 (sqrt(3 x^2 + 4) + sqrt(3) x)) - (4 (3 x^2 + 1) sqrt(3 x^2 + 4))/(3 x^3) + constant
STEP 17
Which is equivalent for restricted x values to: Answer: | | = -(8 (sqrt((3 x^2)/4 + 1) (3 x^2 + 1) - 9/8 sqrt(3) x^3 sinh^(-1)((sqrt(3) x)/2)))/(3 x^3) + constant
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Edited by tododemate® - 2021