What is the integral of (a+bx^3)^2 ?

The integral of (a+bx^3)^2 is a^2x+(abx^4)/2+(b^2x^7)/7+C

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integral of (a+bx^3)^2

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

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

Solution steps

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

Expand

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

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

\int a^{2} d x = a^{2} x

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

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

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

Add a constant to the solution

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

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