What is the integral of x^3(x^2+3)^{10} ?

The integral of x^3(x^2+3)^{10} is

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integral of x^3(x^2+3)^{10}

\int x^{3} (x^{2} + 3)^{10} d x

\frac{1}{264} (11 x^{24} + 360 x^{22} + 5346 x^{20} + 47520 x^{18} + 280665 x^{16} + 1154736 x^{14} + 3367980 x^{12} + 6928416 x^{10} + 9743085 x^{8} + 8660520 x^{6} + 3897234 x^{4}) + C

Solution steps

\int x^{3} (x^{2} + 3)^{10} d x

Apply u-substitution

: \int \frac{u ^{\frac{1}{3}} ( u ^{\frac{2}{3}} + 3 ) ^{10}}{3} d u

= \int \frac{u ^{\frac{1}{3}} ( u ^{\frac{2}{3}} + 3 ) ^{10}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{\frac{1}{3}} (u^{\frac{2}{3}} + 3)^{10} d u

Expand

u^{\frac{1}{3}} (u^{\frac{2}{3}} + 3)^{10} : u^{7} + 30 u^{\frac{19}{3}} + 405 u^{\frac{17}{3}} + 3240 u^{5} + 17010 u^{\frac{13}{3}} + 61236 u^{\frac{11}{3}} + 153090 u^{3} + 262440 u^{\frac{7}{3}} + 295245 u^{\frac{5}{3}} + 196830 u + 59049 u^{\frac{1}{3}}

= \frac{1}{3} \cdot \int u^{7} + 30 u^{\frac{19}{3}} + 405 u^{\frac{17}{3}} + 3240 u^{5} + 17010 u^{\frac{13}{3}} + 61236 u^{\frac{11}{3}} + 153090 u^{3} + 262440 u^{\frac{7}{3}} + 295245 u^{\frac{5}{3}} + 196830 u + 59049 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

= \frac{1}{3} (\int u^{7} d u + \int 30 u^{\frac{19}{3}} d u + \int 405 u^{\frac{17}{3}} d u + \int 3240 u^{5} d u + \int 17010 u^{\frac{13}{3}} d u + \int 61236 u^{\frac{11}{3}} d u + \int 153090 u^{3} d u + \int 262440 u^{\frac{7}{3}} d u + \int 295245 u^{\frac{5}{3}} d u + \int 196830 u d u + \int 59049 u^{\frac{1}{3}} d u)

\int u^{7} d u = \frac{u ^{8}}{8}

\int 30 u^{\frac{19}{3}} d u = \frac{45}{11} u^{\frac{22}{3}}

\int 405 u^{\frac{17}{3}} d u = \frac{243}{4} u^{\frac{20}{3}}

\int 3240 u^{5} d u = 540 u^{6}

\int 17010 u^{\frac{13}{3}} d u = \frac{25515}{8} u^{\frac{16}{3}}

\int 61236 u^{\frac{11}{3}} d u = 13122 u^{\frac{14}{3}}

\int 153090 u^{3} d u = \frac{76545 u ^{4}}{2}

\int 262440 u^{\frac{7}{3}} d u = 78732 u^{\frac{10}{3}}

\int 295245 u^{\frac{5}{3}} d u = \frac{885735}{8} u^{\frac{8}{3}}

\int 196830 u d u = 98415 u^{2}

\int 59049 u^{\frac{1}{3}} d u = \frac{177147}{4} u^{\frac{4}{3}}

= \frac{1}{3} (\frac{u ^{8}}{8} + \frac{45}{11} u^{\frac{22}{3}} + \frac{243}{4} u^{\frac{20}{3}} + 540 u^{6} + \frac{25515}{8} u^{\frac{16}{3}} + 13122 u^{\frac{14}{3}} + \frac{76545 u ^{4}}{2} + 78732 u^{\frac{10}{3}} + \frac{885735}{8} u^{\frac{8}{3}} + 98415 u^{2} + \frac{177147}{4} u^{\frac{4}{3}})

Substitute back

u = x^{3}

= \frac{1}{3} (\frac{( x ^{3} ) ^{8}}{8} + \frac{45}{11} (x^{3})^{\frac{22}{3}} + \frac{243}{4} (x^{3})^{\frac{20}{3}} + 540 (x^{3})^{6} + \frac{25515}{8} (x^{3})^{\frac{16}{3}} + 13122 (x^{3})^{\frac{14}{3}} + \frac{76545 ( x ^{3} ) ^{4}}{2} + 78732 (x^{3})^{\frac{10}{3}} + \frac{885735}{8} (x^{3})^{\frac{8}{3}} + 98415 (x^{3})^{2} + \frac{177147}{4} (x^{3})^{\frac{4}{3}})

Simplify

\frac{1}{3} (\frac{( x ^{3} ) ^{8}}{8} + \frac{45}{11} (x^{3})^{\frac{22}{3}} + \frac{243}{4} (x^{3})^{\frac{20}{3}} + 540 (x^{3})^{6} + \frac{25515}{8} (x^{3})^{\frac{16}{3}} + 13122 (x^{3})^{\frac{14}{3}} + \frac{76545 ( x ^{3} ) ^{4}}{2} + 78732 (x^{3})^{\frac{10}{3}} + \frac{885735}{8} (x^{3})^{\frac{8}{3}} + 98415 (x^{3})^{2} + \frac{177147}{4} (x^{3})^{\frac{4}{3}}) : \frac{1}{264} (11 x^{24} + 360 x^{22} + 5346 x^{20} + 47520 x^{18} + 280665 x^{16} + 1154736 x^{14} + 3367980 x^{12} + 6928416 x^{10} + 9743085 x^{8} + 8660520 x^{6} + 3897234 x^{4})

= \frac{1}{264} (11 x^{24} + 360 x^{22} + 5346 x^{20} + 47520 x^{18} + 280665 x^{16} + 1154736 x^{14} + 3367980 x^{12} + 6928416 x^{10} + 9743085 x^{8} + 8660520 x^{6} + 3897234 x^{4})

Add a constant to the solution

= \frac{1}{264} (11 x^{24} + 360 x^{22} + 5346 x^{20} + 47520 x^{18} + 280665 x^{16} + 1154736 x^{14} + 3367980 x^{12} + 6928416 x^{10} + 9743085 x^{8} + 8660520 x^{6} + 3897234 x^{4}) + C

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