What is the integral of (2a^2x^2-b^3)^2 ?

The integral of (2a^2x^2-b^3)^2 is (4a^4x^5)/5-(4a^2b^3x^3)/3+b^6x+C

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integral of (2a^2x^2-b^3)^2

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

\frac{4 a ^{4} x ^{5}}{5} - \frac{4 a ^{2} b ^{3} x ^{3}}{3} + b^{6} x + C

Solution steps

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

Expand

(2 a^{2} x^{2} - b^{3})^{2} : 4 a^{4} x^{4} - 4 a^{2} b^{3} x^{2} + b^{6}

= \int 4 a^{4} x^{4} - 4 a^{2} b^{3} x^{2} + b^{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 4 a^{4} x^{4} d x - \int 4 a^{2} b^{3} x^{2} d x + \int b^{6} d x

\int 4 a^{4} x^{4} d x = \frac{4 a ^{4} x ^{5}}{5}

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

\int b^{6} d x = b^{6} x

= \frac{4 a ^{4} x ^{5}}{5} - \frac{4 a ^{2} b ^{3} x ^{3}}{3} + b^{6} x

Add a constant to the solution

= \frac{4 a ^{4} x ^{5}}{5} - \frac{4 a ^{2} b ^{3} x ^{3}}{3} + b^{6} x + C

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What is the integral of (2a^2x^2-b^3)^2 ?
The integral of (2a^2x^2-b^3)^2 is (4a^4x^5)/5-(4a^2b^3x^3)/3+b^6x+C