What is the integral of (x^2-18)^8(2x) ?

The integral of (x^2-18)^8(2x) is

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integral of (x^2-18)^8(2x)

\int (x^{2} - 18)^{8} (2 x) d x

x^{2} (x^{2} - 18)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 18)^{7} - \frac{7 x ^{18}}{18} + \frac{189 x ^{16}}{4} - 2430 x^{14} + 68040 x^{12} - 1102248 x^{10} + 9920232 x^{8} - 39680928 x^{6}) + C

Solution steps

\int (x^{2} - 18)^{8} \cdot 2 x d x

Take the constant out:

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

= 2 \cdot \int (x^{2} - 18)^{8} x d x

Apply u-substitution

: \int \frac{( u - 18 ) ^{8}}{2} d u

= 2 \cdot \int \frac{( u - 18 ) ^{8}}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int (u - 18)^{8} d u

Apply Integration By Parts

: (u - 18)^{8} u - \int 8 (u - 18)^{7} u d u

= 2 \cdot \frac{1}{2} ((u - 18)^{8} u - \int 8 (u - 18)^{7} u d u)

\int 8 (u - 18)^{7} u d u = 8 (\frac{1}{2} u^{2} (u - 18)^{7} - \frac{7 u ^{9}}{18} + \frac{189 u ^{8}}{4} - 2430 u^{7} + 68040 u^{6} - 1102248 u^{5} + 9920232 u^{4} - 39680928 u^{3})

= 2 \cdot \frac{1}{2} ((u - 18)^{8} u - 8 (\frac{1}{2} u^{2} (u - 18)^{7} - \frac{7 u ^{9}}{18} + \frac{189 u ^{8}}{4} - 2430 u^{7} + 68040 u^{6} - 1102248 u^{5} + 9920232 u^{4} - 39680928 u^{3}))

Substitute back

u = x^{2}

= 2 \cdot \frac{1}{2} ((x^{2} - 18)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 18)^{7} - \frac{7 ( x ^{2} ) ^{9}}{18} + \frac{189 ( x ^{2} ) ^{8}}{4} - 2430 (x^{2})^{7} + 68040 (x^{2})^{6} - 1102248 (x^{2})^{5} + 9920232 (x^{2})^{4} - 39680928 (x^{2})^{3}))

Simplify

2 \cdot \frac{1}{2} ((x^{2} - 18)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 18)^{7} - \frac{7 ( x ^{2} ) ^{9}}{18} + \frac{189 ( x ^{2} ) ^{8}}{4} - 2430 (x^{2})^{7} + 68040 (x^{2})^{6} - 1102248 (x^{2})^{5} + 9920232 (x^{2})^{4} - 39680928 (x^{2})^{3})) : x^{2} (x^{2} - 18)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 18)^{7} - \frac{7 x ^{18}}{18} + \frac{189 x ^{16}}{4} - 2430 x^{14} + 68040 x^{12} - 1102248 x^{10} + 9920232 x^{8} - 39680928 x^{6})

= x^{2} (x^{2} - 18)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 18)^{7} - \frac{7 x ^{18}}{18} + \frac{189 x ^{16}}{4} - 2430 x^{14} + 68040 x^{12} - 1102248 x^{10} + 9920232 x^{8} - 39680928 x^{6})

Add a constant to the solution

= x^{2} (x^{2} - 18)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 18)^{7} - \frac{7 x ^{18}}{18} + \frac{189 x ^{16}}{4} - 2430 x^{14} + 68040 x^{12} - 1102248 x^{10} + 9920232 x^{8} - 39680928 x^{6}) + C

\frac{d y}{d x}, 3 (x^{2} + y^{2})^{2} = 100 x y

\frac{d}{d x} (\frac{a ^{2} x}{( x + b ) ^{2}})

\int (6 x^{3}) d x

f (x) = \frac{cos ( x )}{cos ( x ) + 1}