Problem to solve
Find the indefinite integral \[\int\frac{x^6}{\sqrt{[x^2+4]^3}} dx\]
Solution
STEP 1
Take the integral: integral x^6/(x^2 + 4)^(3/2) dx
STEP 2
For the integrand x^6/(x^2 + 4)^(3/2), substitute x = 2 tan(u) and dx = 2 sec^2(u) du. Then (x^2 + 4)^(3/2) = (4 tan^2(u) + 4)^(3/2) = 8 sec^3(u) and u = tan^(-1)(x/2): = 2 integral8 sin(u) tan^5(u) du
STEP 3
Factor out constants: = 16 integral sin(u) tan^5(u) du
STEP 4
For the integrand sin(u) tan^5(u), use the trigonometric identity tan^2(u) = sec^2(u) - 1: = 16 integral sin(u) tan(u) (sec^2(u) - 1)^2 du
STEP 5
Expanding the integrand sin(u) tan(u) (sec^2(u) - 1)^2 gives sin(u) tan(u) + tan^2(u) sec^3(u) - 2 tan^2(u) sec(u): = 16 integral(sin(u) tan(u) + tan^2(u) sec^3(u) - 2 tan^2(u) sec(u)) du
STEP 6
Integrate the sum term by term and factor out constants: = 16 integral tan^2(u) sec^3(u) du - 32 integral tan^2(u) sec(u) du + 16 integral sin(u) tan(u) du
STEP 7
For the integrand tan^2(u) sec^3(u), write tan^2(u) as sec^2(u) - 1: = 16 integral sec^3(u) (sec^2(u) - 1) du - 32 integral tan^2(u) sec(u) du + 16 integral sin(u) tan(u) du
STEP 8
Expanding the integrand sec^3(u) (sec^2(u) - 1) gives sec^5(u) - sec^3(u): = 16 integral(sec^5(u) - sec^3(u)) du - 32 integral tan^2(u) sec(u) du + 16 integral sin(u) tan(u) du
STEP 9
Integrate the sum term by term and factor out constants: = 16 integral sec^5(u) du - 16 integral sec^3(u) du - 32 integral tan^2(u) sec(u) du + 16 integral sin(u) tan(u) du
STEP 10
Use the reduction formula, integral sec^m(u) du = (sin(u) sec^(m - 1)(u))/(m - 1) + (m - 2)/(m - 1) integral sec^(-2 + m)(u) du, where m = 5: = 4 tan(u) sec^3(u) - 4 integral sec^3(u) du - 32 integral tan^2(u) sec(u) du + 16 integral sin(u) tan(u) du
STEP 11
Use the reduction formula, integral sec^m(u) du = (sin(u) sec^(m - 1)(u))/(m - 1) + (m - 2)/(m - 1) integral sec^(-2 + m)(u) du, where m = 3: = -2 tan(u) sec(u) + 4 tan(u) sec^3(u) - 2 integral sec(u) du - 32 integral tan^2(u) sec(u) du + 16 integral sin(u) tan(u) du
STEP 12
The integral of sec(u) is log(tan(u) + sec(u)): = -2 log(tan(u) + sec(u)) - 2 tan(u) sec(u) + 4 tan(u) sec^3(u) - 32 integral tan^2(u) sec(u) du + 16 integral sin(u) tan(u) du
STEP 13
For the integrand tan^2(u) sec(u), write tan^2(u) as sec^2(u) - 1: = -2 log(tan(u) + sec(u)) - 2 tan(u) sec(u) + 4 tan(u) sec^3(u) - 32 integral sec(u) (sec^2(u) - 1) du + 16 integral sin(u) tan(u) du
STEP 14
Expanding the integrand sec(u) (sec^2(u) - 1) gives sec^3(u) - sec(u): = -2 log(tan(u) + sec(u)) - 2 tan(u) sec(u) + 4 tan(u) sec^3(u) - 32 integral(sec^3(u) - sec(u)) du + 16 integral sin(u) tan(u) du
STEP 15
Integrate the sum term by term and factor out constants: = -2 log(tan(u) + sec(u)) - 2 tan(u) sec(u) + 4 tan(u) sec^3(u) - 32 integral sec^3(u) du + 32 integral sec(u) du + 16 integral sin(u) tan(u) du
STEP 16
Use the reduction formula, integral sec^m(u) du = (sin(u) sec^(m - 1)(u))/(m - 1) + (m - 2)/(m - 1) integral sec^(-2 + m)(u) du, where m = 3: = -2 log(tan(u) + sec(u)) - 18 tan(u) sec(u) + 4 tan(u) sec^3(u) + 16 integral sec(u) du + 16 integral sin(u) tan(u) du
STEP 17
The integral of sec(u) is log(tan(u) + sec(u)): = 14 log(tan(u) + sec(u)) - 18 tan(u) sec(u) + 4 tan(u) sec^3(u) + 16 integral sin(u) tan(u) du
STEP 18
For the integrand sin(u) tan(u), use the trigonometric identities sin^2(u) + cos^2(u) = 1 and tan(u) = sin(u)/cos(u): = 14 log(tan(u) + sec(u)) - 18 tan(u) sec(u) + 4 tan(u) sec^3(u) + 16 integral(sec(u) - cos(u)) du
STEP 19
Integrate the sum term by term and factor out constants: = 14 log(tan(u) + sec(u)) - 18 tan(u) sec(u) + 4 tan(u) sec^3(u) + 16 integral sec(u) du - 16 integral cos(u) du
STEP 20
The integral of sec(u) is log(tan(u) + sec(u)): = 30 log(tan(u) + sec(u)) - 18 tan(u) sec(u) + 4 tan(u) sec^3(u) - 16 integral cos(u) du
STEP 21
The integral of cos(u) is sin(u): = -16 sin(u) + 4 tan(u) sec^3(u) - 18 tan(u) sec(u) + 30 log(tan(u) + sec(u)) + constant
STEP 22
Substitute back for u = tan^(-1)(x/2): = -16 sin(tan^(-1)(x/2)) + 4 tan(tan^(-1)(x/2)) sec(tan^(-1)(x/2))^3 - 18 tan(tan^(-1)(x/2)) sec(tan^(-1)(x/2)) + 30 log(tan(tan^(-1)(x/2)) + sec(tan^(-1)(x/2))) + constant
STEP 23
Simplify using sec(tan^(-1)(z)) = sqrt(z^2 + 1), sin(tan^(-1)(z)) = z/sqrt(z^2 + 1) and tan(tan^(-1)(z)) = z: = 1/4 x (x^2 + 4)^(3/2) - 9/2 x sqrt(x^2 + 4) - (16 x)/sqrt(x^2 + 4) + 30 log(1/2 (sqrt(x^2 + 4) + x)) + constant
STEP 24
Factor the answer a different way: = (x^5 - 10 x^3 + 120 sqrt(x^2 + 4) log(1/2 (sqrt(x^2 + 4) + x)) - 120 x)/(4 sqrt(x^2 + 4)) + constant
STEP 25
Which is equivalent for restricted x values to: Answer: | | = (x^5 - 10 x^3 + 120 sqrt(x^2 + 4) sinh^(-1)(x/2) - 120 x)/(4 sqrt(x^2 + 4)) + constant
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Edited by tododemate® - 2021