Problem to solve
Find the indefinite integral \[\int[12x^{15}-3x^7-4x^3]\cdot e^{2x^4} dx\]
Solution
STEP 1
Take the integral: integral e^(2 x^4) (12 x^15 - 3 x^7 - 4 x^3) dx
STEP 2
Expanding the integrand e^(2 x^4) (12 x^15 - 3 x^7 - 4 x^3) gives 12 e^(2 x^4) x^15 - 3 e^(2 x^4) x^7 - 4 e^(2 x^4) x^3: = integral(12 e^(2 x^4) x^15 - 3 e^(2 x^4) x^7 - 4 e^(2 x^4) x^3) dx
STEP 3
Integrate the sum term by term and factor out constants: = 12 integral e^(2 x^4) x^15 dx - 3 integral e^(2 x^4) x^7 dx - 4 integral e^(2 x^4) x^3 dx
STEP 4
For the integrand e^(2 x^4) x^15, substitute u = x^4 and du = 4 x^3 dx: = 3 integral e^(2 u) u^3 du - 3 integral e^(2 x^4) x^7 dx - 4 integral e^(2 x^4) x^3 dx
STEP 5
For the integrand e^(2 u) u^3, integrate by parts, integral f dg = f g - integral g df, where f = u^3, dg = e^(2 u) du, df = 3 u^2 du, g = e^(2 u)/2: = 3/2 e^(2 u) u^3 - 9/2 integral e^(2 u) u^2 du - 3 integral e^(2 x^4) x^7 dx - 4 integral e^(2 x^4) x^3 dx
STEP 6
For the integrand e^(2 u) u^2, integrate by parts, integral f dg = f g - integral g df, where f = u^2, dg = e^(2 u) du, df = 2 u du, g = e^(2 u)/2: = -9/4 e^(2 u) u^2 + 3/2 e^(2 u) u^3 + 9/2 integral e^(2 u) u du - 3 integral e^(2 x^4) x^7 dx - 4 integral e^(2 x^4) x^3 dx
STEP 7
For the integrand e^(2 u) u, integrate by parts, integral f dg = f g - integral g df, where f = u, dg = e^(2 u) du, df = du, g = e^(2 u)/2: = 9/4 e^(2 u) u - 9/4 e^(2 u) u^2 + 3/2 e^(2 u) u^3 - 9/4 integral e^(2 u) du - 3 integral e^(2 x^4) x^7 dx - 4 integral e^(2 x^4) x^3 dx
STEP 8
The integral of e^(2 u) is e^(2 u)/2: = -(9 e^(2 u))/8 + 9/4 e^(2 u) u - 9/4 e^(2 u) u^2 + 3/2 e^(2 u) u^3 - 3 integral e^(2 x^4) x^7 dx - 4 integral e^(2 x^4) x^3 dx
STEP 9
For the integrand e^(2 x^4) x^7, substitute s = x^4 and ds = 4 x^3 dx: = -(9 e^(2 u))/8 + 9/4 e^(2 u) u - 9/4 e^(2 u) u^2 + 3/2 e^(2 u) u^3 - 3/4 integral e^(2 s) s ds - 4 integral e^(2 x^4) x^3 dx
STEP 10
For the integrand e^(2 s) s, integrate by parts, integral f dg = f g - integral g df, where f = s, dg = e^(2 s) ds, df = ds, g = e^(2 s)/2: = -(9 e^(2 u))/8 - 3/8 e^(2 s) s + 9/4 e^(2 u) u - 9/4 e^(2 u) u^2 + 3/2 e^(2 u) u^3 + 3/8 integral e^(2 s) ds - 4 integral e^(2 x^4) x^3 dx
STEP 11
The integral of e^(2 s) is e^(2 s)/2: = (3 e^(2 s))/16 - (9 e^(2 u))/8 - 3/8 e^(2 s) s + 9/4 e^(2 u) u - 9/4 e^(2 u) u^2 + 3/2 e^(2 u) u^3 - 4 integral e^(2 x^4) x^3 dx
STEP 12
For the integrand e^(2 x^4) x^3, substitute p = 2 x^4 and dp = 8 x^3 dx: = (3 e^(2 s))/16 - (9 e^(2 u))/8 - 3/8 e^(2 s) s + 9/4 e^(2 u) u - 9/4 e^(2 u) u^2 + 3/2 e^(2 u) u^3 - 1/2 integral e^p dp
STEP 13
The integral of e^p is e^p: = -e^p/2 + (3 e^(2 s))/16 - 3/8 e^(2 s) s + 3/2 e^(2 u) u^3 - 9/4 e^(2 u) u^2 + 9/4 e^(2 u) u - (9 e^(2 u))/8 + constant
STEP 14
Substitute back for p = 2 x^4: = (3 e^(2 s))/16 - 3/8 e^(2 s) s + 3/2 e^(2 u) u^3 - 9/4 e^(2 u) u^2 + 9/4 e^(2 u) u - (9 e^(2 u))/8 - e^(2 x^4)/2 + constant
STEP 15
Substitute back for s = x^4: = 3/2 e^(2 u) u^3 - 9/4 e^(2 u) u^2 - (9 e^(2 u))/8 + 9/4 e^(2 u) u - 3/8 e^(2 x^4) x^4 - (5 e^(2 x^4))/16 + constant
STEP 16
Substitute back for u = x^4: = 15/8 e^(2 x^4) x^4 - (23 e^(2 x^4))/16 + 3/2 e^(2 x^4) x^12 - 9/4 e^(2 x^4) x^8 + constant
STEP 17
Which is equal to: Answer: | | = 1/16 e^(2 x^4) (24 x^12 - 36 x^8 + 30 x^4 - 23) + constant
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Edited by tododemate® - 2021