Problem to solve
Find the indefinite integral \[\int[-4x^4+6x^2+8x-1]\cos[x] dx\]
Solution
STEP 1
Take the integral: integral(-4 x^4 + 6 x^2 + 8 x - 1) cos(x) dx
STEP 2
Expanding the integrand (-4 x^4 + 6 x^2 + 8 x - 1) cos(x) gives -4 x^4 cos(x) + 6 x^2 cos(x) + 8 x cos(x) - cos(x): = integral(-4 x^4 cos(x) + 6 x^2 cos(x) + 8 x cos(x) - cos(x)) dx
STEP 3
Integrate the sum term by term and factor out constants: = -4 integral x^4 cos(x) dx + 6 integral x^2 cos(x) dx + 8 integral x cos(x) dx - integral cos(x) dx
STEP 4
For the integrand x^4 cos(x), integrate by parts, integral f dg = f g - integral g df, where f = x^4, dg = cos(x) dx, df = 4 x^3 dx, g = sin(x): = -4 x^4 sin(x) + 16 integral x^3 sin(x) dx + 6 integral x^2 cos(x) dx + 8 integral x cos(x) dx - integral cos(x) dx
STEP 5
For the integrand x^3 sin(x), integrate by parts, integral f dg = f g - integral g df, where f = x^3, dg = sin(x) dx, df = 3 x^2 dx, g = -cos(x): = -16 x^3 cos(x) - 4 x^4 sin(x) + 54 integral x^2 cos(x) dx + 8 integral x cos(x) dx - integral cos(x) dx
STEP 6
For the integrand x^2 cos(x), integrate by parts, integral f dg = f g - integral g df, where f = x^2, dg = cos(x) dx, df = 2 x dx, g = sin(x): = -16 x^3 cos(x) + 54 x^2 sin(x) - 4 x^4 sin(x) - 108 integral x sin(x) dx + 8 integral x cos(x) dx - integral cos(x) dx
STEP 7
For the integrand x sin(x), integrate by parts, integral f dg = f g - integral g df, where f = x, dg = sin(x) dx, df = dx, g = -cos(x): = 108 x cos(x) - 16 x^3 cos(x) + 54 x^2 sin(x) - 4 x^4 sin(x) - 109 integral cos(x) dx + 8 integral x cos(x) dx
STEP 8
The integral of cos(x) is sin(x): = 108 x cos(x) - 16 x^3 cos(x) - 109 sin(x) + 54 x^2 sin(x) - 4 x^4 sin(x) + 8 integral x cos(x) dx
STEP 9
For the integrand x cos(x), integrate by parts, integral f dg = f g - integral g df, where f = x, dg = cos(x) dx, df = dx, g = sin(x): = 108 x cos(x) - 16 x^3 cos(x) - 109 sin(x) + 8 x sin(x) + 54 x^2 sin(x) - 4 x^4 sin(x) - 8 integral sin(x) dx
STEP 10
The integral of sin(x) is -cos(x): = -4 x^4 sin(x) - 16 x^3 cos(x) + 54 x^2 sin(x) + 8 x sin(x) - 109 sin(x) + 108 x cos(x) + 8 cos(x) + constant
STEP 11
Which is equal to: Answer: | | = (-16 x^3 + 108 x + 8) cos(x) + (-4 x^4 + 54 x^2 + 8 x - 109) sin(x) + constant
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Edited by tododemate® - 2021