What is the integral of (9x(x^2-10)^8)/5 ?

The integral of (9x(x^2-10)^8)/5 is

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integral of (9x(x^2-10)^8)/5

\int \frac{9 x ( x ^{2} - 10 ) ^{8}}{5} d x

\frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6}))) + C

Solution steps

\int \frac{9 x ( x ^{2} - 10 ) ^{8}}{5} d x

Take the constant out:

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

= \frac{9}{5} \cdot \int x (x^{2} - 10)^{8} d x

Apply u-substitution

: \int \frac{( u - 10 ) ^{8}}{2} d u

= \frac{9}{5} \cdot \int \frac{( u - 10 ) ^{8}}{2} d u

Take the constant out:

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

= \frac{9}{5} \cdot \frac{1}{2} \cdot \int (u - 10)^{8} d u

Apply Integration By Parts

: (u - 10)^{8} u - \int 8 (u - 10)^{7} u d u

= \frac{9}{5} \cdot \frac{1}{2} ((u - 10)^{8} u - \int 8 (u - 10)^{7} u d u)

\int 8 (u - 10)^{7} u d u = 8 (\frac{1}{2} u^{2} (u - 10)^{7} - \frac{1}{36} (14 u^{9} - 945 u^{8} + 27000 u^{7} - 420000 u^{6} + 3780000 u^{5} - 18900000 u^{4} + 42000000 u^{3}))

= \frac{9}{5} \cdot \frac{1}{2} ((u - 10)^{8} u - 8 (\frac{1}{2} u^{2} (u - 10)^{7} - \frac{1}{36} (14 u^{9} - 945 u^{8} + 27000 u^{7} - 420000 u^{6} + 3780000 u^{5} - 18900000 u^{4} + 42000000 u^{3})))

Substitute back

u = x^{2}

= \frac{9}{5} \cdot \frac{1}{2} ((x^{2} - 10)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 10)^{7} - \frac{1}{36} (14 (x^{2})^{9} - 945 (x^{2})^{8} + 27000 (x^{2})^{7} - 420000 (x^{2})^{6} + 3780000 (x^{2})^{5} - 18900000 (x^{2})^{4} + 42000000 (x^{2})^{3})))

Simplify

\frac{9}{5} \cdot \frac{1}{2} ((x^{2} - 10)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 10)^{7} - \frac{1}{36} (14 (x^{2})^{9} - 945 (x^{2})^{8} + 27000 (x^{2})^{7} - 420000 (x^{2})^{6} + 3780000 (x^{2})^{5} - 18900000 (x^{2})^{4} + 42000000 (x^{2})^{3}))) : \frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6})))

= \frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6})))

Add a constant to the solution

= \frac{9}{10} (x^{2} (x^{2} - 10)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 10)^{7} - \frac{1}{36} (14 x^{18} - 945 x^{16} + 27000 x^{14} - 420000 x^{12} + 3780000 x^{10} - 18900000 x^{8} + 42000000 x^{6}))) + C

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