What is the integral of 12x^3(3x^4+4)^4 ?

The integral of 12x^3(3x^4+4)^4 is 4(216x^{12}+243/20 x^{20}+81x^{16}+288x^8+192x^4)+C

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integral of 12x^3(3x^4+4)^4

\int 12 x^{3} (3 x^{4} + 4)^{4} d x

4 (216 x^{12} + \frac{243}{20} x^{20} + 81 x^{16} + 288 x^{8} + 192 x^{4}) + C

Solution steps

\int 12 x^{3} (3 x^{4} + 4)^{4} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 12 \cdot \int x^{3} (3 x^{4} + 4)^{4} d x

Apply u-substitution

= 12 \cdot \int \frac{u ^{\frac{1}{3}} ( 3 u ^{\frac{4}{3}} + 4 ) ^{4}}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 12 \cdot \frac{1}{3} \cdot \int u^{\frac{1}{3}} (3 u^{\frac{4}{3}} + 4)^{4} d u

Expand

u^{\frac{1}{3}} (3 u^{\frac{4}{3}} + 4)^{4} : 81 u^{\frac{17}{3}} + 432 u^{\frac{13}{3}} + 864 u^{3} + 768 u^{\frac{5}{3}} + 256 u^{\frac{1}{3}}

= 12 \cdot \frac{1}{3} \cdot \int 81 u^{\frac{17}{3}} + 432 u^{\frac{13}{3}} + 864 u^{3} + 768 u^{\frac{5}{3}} + 256 u^{\frac{1}{3}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 12 \cdot \frac{1}{3} (\int 81 u^{\frac{17}{3}} d u + \int 432 u^{\frac{13}{3}} d u + \int 864 u^{3} d u + \int 768 u^{\frac{5}{3}} d u + \int 256 u^{\frac{1}{3}} d u)

\int 81 u^{\frac{17}{3}} d u = \frac{243}{20} u^{\frac{20}{3}}

\int 432 u^{\frac{13}{3}} d u = 81 u^{\frac{16}{3}}

\int 864 u^{3} d u = 216 u^{4}

\int 768 u^{\frac{5}{3}} d u = 288 u^{\frac{8}{3}}

\int 256 u^{\frac{1}{3}} d u = 192 u^{\frac{4}{3}}

= 12 \cdot \frac{1}{3} (\frac{243}{20} u^{\frac{20}{3}} + 81 u^{\frac{16}{3}} + 216 u^{4} + 288 u^{\frac{8}{3}} + 192 u^{\frac{4}{3}})

Substitute back

u = x^{3}

= 12 \cdot \frac{1}{3} (\frac{243}{20} (x^{3})^{\frac{20}{3}} + 81 (x^{3})^{\frac{16}{3}} + 216 (x^{3})^{4} + 288 (x^{3})^{\frac{8}{3}} + 192 (x^{3})^{\frac{4}{3}})

Simplify

12 \cdot \frac{1}{3} (\frac{243}{20} (x^{3})^{\frac{20}{3}} + 81 (x^{3})^{\frac{16}{3}} + 216 (x^{3})^{4} + 288 (x^{3})^{\frac{8}{3}} + 192 (x^{3})^{\frac{4}{3}}) : 4 (216 x^{12} + \frac{243}{20} x^{20} + 81 x^{16} + 288 x^{8} + 192 x^{4})

= 4 (216 x^{12} + \frac{243}{20} x^{20} + 81 x^{16} + 288 x^{8} + 192 x^{4})

Add a constant to the solution

= 4 (216 x^{12} + \frac{243}{20} x^{20} + 81 x^{16} + 288 x^{8} + 192 x^{4}) + C

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What is the integral of 12x^3(3x^4+4)^4 ?
The integral of 12x^3(3x^4+4)^4 is 4(216x^{12}+243/20 x^{20}+81x^{16}+288x^8+192x^4)+C