What is the integral of (3x^2+1)^6 ?

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integral of (3x^2+1)^6

\int (3 x^{2} + 1)^{6} d x

\frac{1}{3} (3 x (3 x^{2} + 1)^{6} - 12 (\frac{729 3 x ^{13}}{13} + \frac{1215 3 x ^{11}}{11} + 903 x^{9} + \frac{270 3 x ^{7}}{7} + 93 x^{5} + 3 x^{3})) + C

Solution steps

\int (3 x^{2} + 1)^{6} d x

Apply Integral Substitution

= \int \frac{( u ^{2} + 1 ) ^{6}}{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^{2} + 1)^{6} d u

Apply Integration By Parts

= \frac{1}{3} (u (u^{2} + 1)^{6} - \int 12 u^{2} (u^{2} + 1)^{5} d u)

\int 12 u^{2} (u^{2} + 1)^{5} d u = 12 (\frac{u ^{13}}{13} + \frac{5 u ^{11}}{11} + \frac{10 u ^{9}}{9} + \frac{10 u ^{7}}{7} + u^{5} + \frac{u ^{3}}{3})

= \frac{1}{3} (u (u^{2} + 1)^{6} - 12 (\frac{u ^{13}}{13} + \frac{5 u ^{11}}{11} + \frac{10 u ^{9}}{9} + \frac{10 u ^{7}}{7} + u^{5} + \frac{u ^{3}}{3}))

Substitute back

u = 3 x

= \frac{1}{3} (3 x ((3 x)^{2} + 1)^{6} - 12 (\frac{( 3 x ) ^{13}}{13} + \frac{5 ( 3 x ) ^{11}}{11} + \frac{10 ( 3 x ) ^{9}}{9} + \frac{10 ( 3 x ) ^{7}}{7} + (3 x)^{5} + \frac{( 3 x ) ^{3}}{3}))

Simplify

\frac{1}{3} (3 x ((3 x)^{2} + 1)^{6} - 12 (\frac{( 3 x ) ^{13}}{13} + \frac{5 ( 3 x ) ^{11}}{11} + \frac{10 ( 3 x ) ^{9}}{9} + \frac{10 ( 3 x ) ^{7}}{7} + (3 x)^{5} + \frac{( 3 x ) ^{3}}{3})) : \frac{1}{3} (3 x (3 x^{2} + 1)^{6} - 12 (\frac{729 3 x ^{13}}{13} + \frac{1215 3 x ^{11}}{11} + 903 x^{9} + \frac{270 3 x ^{7}}{7} + 93 x^{5} + 3 x^{3}))

= \frac{1}{3} (3 x (3 x^{2} + 1)^{6} - 12 (\frac{729 3 x ^{13}}{13} + \frac{1215 3 x ^{11}}{11} + 903 x^{9} + \frac{270 3 x ^{7}}{7} + 93 x^{5} + 3 x^{3}))

Add a constant to the solution

= \frac{1}{3} (3 x (3 x^{2} + 1)^{6} - 12 (\frac{729 3 x ^{13}}{13} + \frac{1215 3 x ^{11}}{11} + 903 x^{9} + \frac{270 3 x ^{7}}{7} + 93 x^{5} + 3 x^{3})) + C