Verified answer

∫ 3x^5 * e^x dx ===> by parts

u = 3x^5. .. . . .. . .dv = e^x

du = 15x^4. . .. . . v =e^x

u * v - ∫ v * du

3x^5 * e^x - ∫ e^x * 15x^4

3x^5 * e^x - ∫ 15x^4 * e^x ===> by parts

u = 15x^4. .. . . .. . .dv = e^x

du = 60x^3. . .. . . v =e^x

u * v - ∫ v * du

3x^5 * e^x - [ 15x^4 * e^x - ∫ 60x^3 * e^x ]

3x^5 * e^x - 15x^4 * e^x + ∫ 60x^3 * e^x ===> by parts

u = 60x^3. .. . . .. . .dv = e^x

du = 180x^2. . .. . . v =e^x

u * v - ∫ v * du

3x^5 * e^x - 15x^4 * e^x + [ 60x^3 * e^x - ∫ 180x^2 * e^x ]

3x^5 * e^x - 15x^4 * e^x + 60x^3 * e^x - ∫ 180x^2 * e^x ==> by parts

u = 180x^2. .. . . .. . .dv = e^x

du = 360x. . .. . . v =e^x

u * v - ∫ v * du

3x^5 * e^x - 15x^4 * e^x + 60x^3 * e^x - [ 180x^2 * e^x - ∫ 360x * e^x ]

3x^5 * e^x - 15x^4 * e^x + 60x^3 * e^x - 180x^2 * e^x + ∫ 360x * e^x ==> by parts

u = 360x. .. . . .. . .dv = e^x

du = 360. . .. . . v =e^x

u * v - ∫ v * du

3x^5 * e^x - 15x^4 * e^x + 60x^3 * e^x - 180x^2 * e^x + [ 360x * e^x - ∫ 360 * e^x ]

3x^5 * e^x - 15x^4 * e^x + 60x^3 * e^x - 180x^2 * e^x + 360x * e^x - 360 * e^x + C

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can please the discovery that i found in my homepage, which is as the following:

if we have as:

∫ x e^x dx ====> e^x [ x - 1 ] + C

∫ x^2 e^x dx ====> e^x [ x^2 - 2x + 2 ] + C

∫ x^3 e^x dx ====> e^x [ x^3 - 3x^2 + 6x - 6 ] + C

our example as :

∫ 3 * x^5 e^x dx ====> 3e^x [ x^5 - 5x^4 + 20x^3 - 60x^2 + 120x - 120 ] + C

which it will be as :

3 * [ x^5 * e^x - 5x^4 * e^x + 20x^3 * e^x - 60x^2 * e^x + 120x * - 120e^x ] + C

3x^5 * e^x - 15x^4 * e^x + 60x^3 * e^x - 180x^2 * e^x + 360x * - 360e^x + C

A=e^x(3x^5)-integral(15x^4 e^x dx)=

(e^x)(3x^5)-B

integral(15x^4 e^x dx)=B

B=e^x(15x^4)-integral(e^x)(60x^3)dx

integral(e^x)(60x^3)dx=C

C=60(e^x)(x^3)-D

D=integral (180x^2)e^xdx

D=(180x^2)e^x-E

E=integral (360x)e^x dx

E=(360x)e^x-F

F=integral360e^x=360e^x

so E=(360e^x)(x-1)

D=(180e^x)(x^2-2x-2)

C=(60e^x)(x^3-3x^2-6x-6)

B=(15e^x)(x^4-4X^3-12x^2-24x-24

A=(3e^x)(x^5-5x^4-20x^3-60x^2-120x-120)

A+c is the answer!

Integrate the original integrand by parts:

∫ 3x⁵℮˟ dx

Let f'(x) = ℮˟

f(x) = ℮˟

Let g(x) = 3x⁵

g'(x) = 15x⁴

∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g'(x) dx

∫ 3x⁵℮˟ dx = 3x⁵℮˟ - ∫ 15x⁴℮˟ dx

Integrate the new integrand by parts:

∫ 15x⁴℮˟ dx

Let f'(x) = ℮˟

f(x) = ℮˟

Let g(x) = 15x⁴

g'(x) = 60x³

∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g'(x) dx

∫ 15x⁴℮˟ dx = 15x⁴℮˟ - ∫ 60x³℮˟ dx

Integrate the new integrand by parts:

∫ 60x³℮˟ dx

Let f'(x) = ℮˟

f(x) = ℮˟

Let g(x) = 60x³

g'(x) = 180x²

∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g'(x) dx

∫ 60x³℮˟ dx = 60x³℮˟ - ∫ 180x²℮˟ dx

Integrate the new integrand by parts:

∫ 180x²℮˟ dx

Let f'(x) = ℮˟

f(x) = ℮˟

Let g(x) = 180x²

g'(x) = 360x

∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g'(x) dx

∫ 180x²℮˟ dx = 180x²℮˟ - ∫ 360x℮˟ dx

Integrate the new integrand by parts:

∫ 360x℮˟ dx

Let f'(x) = ℮˟

f(x) = ℮˟

Let g(x) = 360x

g'(x) = 360

∫ f'(x)g(x) dx = f(x)g(x) - ∫ f(x)g'(x) dx

∫ 360x℮˟ dx = 360x℮˟ - ∫ 360℮˟ dx

∫ 360x℮˟ dx = 360x℮˟ - 360℮˟

Put it all together to find the original integral:

∫ 3x⁵℮˟ dx = 3x⁵℮˟ - ∫ 15x⁴℮˟ dx

∫ 3x⁵℮˟ dx = 3x⁵℮˟ - (15x⁴℮˟ - ∫ 60x³℮˟ dx)

∫ 3x⁵℮˟ dx = 3x⁵℮˟ - 15x⁴℮˟ + 60x³℮˟ - ∫ 180x²℮˟ dx

∫ 3x⁵℮˟ dx = 3x⁵℮˟ - 15x⁴℮˟ + 60x³℮˟ - (180x²℮˟ - ∫ 360x℮˟ dx)

∫ 3x⁵℮˟ dx = 3x⁵℮˟ - 15x⁴℮˟ + 60x³℮˟ - 180x²℮˟ + ∫ 360x℮˟ dx

∫ 3x⁵℮˟ dx = 3x⁵℮˟ - 15x⁴℮˟ + 60x³℮˟ - 180x²℮˟ + 360x℮˟ - 360℮˟ + C

∫ 3x⁵℮˟ dx = (3x⁵ - 15x⁴ + 60x³ - 180x² + 360x - 360)℮˟ + C