What is the integral of x^2(5x^3+7x^2-6x-3) ?

The integral of x^2(5x^3+7x^2-6x-3) is (5x^6)/6+(7x^5)/5-(3x^4)/2-x^3+C

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integral of x^2(5x^3+7x^2-6x-3)

\int x^{2} (5 x^{3} + 7 x^{2} - 6 x - 3) d x

\frac{5 x ^{6}}{6} + \frac{7 x ^{5}}{5} - \frac{3 x ^{4}}{2} - x^{3} + C

Solution steps

\int x^{2} (5 x^{3} + 7 x^{2} - 6 x - 3) d x

Expand

x^{2} (5 x^{3} + 7 x^{2} - 6 x - 3) : 5 x^{5} + 7 x^{4} - 6 x^{3} - 3 x^{2}

= \int 5 x^{5} + 7 x^{4} - 6 x^{3} - 3 x^{2} 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 5 x^{5} d x + \int 7 x^{4} d x - \int 6 x^{3} d x - \int 3 x^{2} d x

\int 5 x^{5} d x = \frac{5 x ^{6}}{6}

\int 7 x^{4} d x = \frac{7 x ^{5}}{5}

\int 6 x^{3} d x = \frac{3 x ^{4}}{2}

\int 3 x^{2} d x = x^{3}

= \frac{5 x ^{6}}{6} + \frac{7 x ^{5}}{5} - \frac{3 x ^{4}}{2} - x^{3}

Add a constant to the solution

= \frac{5 x ^{6}}{6} + \frac{7 x ^{5}}{5} - \frac{3 x ^{4}}{2} - x^{3} + C

sin (2 x) d x + cos (3 y) d y = 0, y (\frac{π}{2}) = \frac{π}{3}

\int \frac{4}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x

x \to + 3 lim (4 - x)

x \to 0 lim (x + x)

\frac{d}{d x} (c x^{2})

What is the integral of x^2(5x^3+7x^2-6x-3) ?
The integral of x^2(5x^3+7x^2-6x-3) is (5x^6)/6+(7x^5)/5-(3x^4)/2-x^3+C