What is the integral of (a+bx^3)^3 ?

The integral of (a+bx^3)^3 is a^3x+(3a^2bx^4)/4+(3ab^2x^7)/7+(b^3x^{10})/(10)+C

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integral of (a+bx^3)^3

\int (a + b x^{3})^{3} d x

a^{3} x + \frac{3 a ^{2} b x ^{4}}{4} + \frac{3 a b ^{2} x ^{7}}{7} + \frac{b ^{3} x ^{10}}{10} + C

Solution steps

\int (a + b x^{3})^{3} d x

Expand

(a + b x^{3})^{3} : a^{3} + 3 a^{2} b x^{3} + 3 a b^{2} x^{6} + b^{3} x^{9}

= \int a^{3} + 3 a^{2} b x^{3} + 3 a b^{2} x^{6} + b^{3} x^{9} d x

Apply the Sum Rule:

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

= \int a^{3} d x + \int 3 a^{2} b x^{3} d x + \int 3 a b^{2} x^{6} d x + \int b^{3} x^{9} d x

\int a^{3} d x = a^{3} x

\int 3 a^{2} b x^{3} d x = \frac{3 a ^{2} b x ^{4}}{4}

\int 3 a b^{2} x^{6} d x = \frac{3 a b ^{2} x ^{7}}{7}

\int b^{3} x^{9} d x = \frac{b ^{3} x ^{10}}{10}

= a^{3} x + \frac{3 a ^{2} b x ^{4}}{4} + \frac{3 a b ^{2} x ^{7}}{7} + \frac{b ^{3} x ^{10}}{10}

Add a constant to the solution

= a^{3} x + \frac{3 a ^{2} b x ^{4}}{4} + \frac{3 a b ^{2} x ^{7}}{7} + \frac{b ^{3} x ^{10}}{10} + C

\frac{d}{d x} (2 x - 3 sin (x))

x \to 3 lim (- 2 x^{3})

\int \frac{( x ^{2} - 16 )}{x ^{3}} d x

\frac{1}{x + 4}

\int \frac{20}{( x - 1 ) ( x ^{2} + 9 )} d x

What is the integral of (a+bx^3)^3 ?
The integral of (a+bx^3)^3 is a^3x+(3a^2bx^4)/4+(3ab^2x^7)/7+(b^3x^{10})/(10)+C