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

The integral of x^3(3+x^2)^5 is (243x^4)/4+(135x^6)/2+(135x^8)/4+9x^{10}+(5x^{12})/4+(x^{14})/(14)+C

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integral of x^3(3+x^2)^5

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

\frac{243 x ^{4}}{4} + \frac{135 x ^{6}}{2} + \frac{135 x ^{8}}{4} + 9 x^{10} + \frac{5 x ^{12}}{4} + \frac{x ^{14}}{14} + C

Solution steps

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

Expand

x^{3} (3 + x^{2})^{5} : 243 x^{3} + 405 x^{5} + 270 x^{7} + 90 x^{9} + 15 x^{11} + x^{13}

= \int 243 x^{3} + 405 x^{5} + 270 x^{7} + 90 x^{9} + 15 x^{11} + x^{13} 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 243 x^{3} d x + \int 405 x^{5} d x + \int 270 x^{7} d x + \int 90 x^{9} d x + \int 15 x^{11} d x + \int x^{13} d x

\int 243 x^{3} d x = \frac{243 x ^{4}}{4}

\int 405 x^{5} d x = \frac{135 x ^{6}}{2}

\int 270 x^{7} d x = \frac{135 x ^{8}}{4}

\int 90 x^{9} d x = 9 x^{10}

\int 15 x^{11} d x = \frac{5 x ^{12}}{4}

\int x^{13} d x = \frac{x ^{14}}{14}

= \frac{243 x ^{4}}{4} + \frac{135 x ^{6}}{2} + \frac{135 x ^{8}}{4} + 9 x^{10} + \frac{5 x ^{12}}{4} + \frac{x ^{14}}{14}

Add a constant to the solution

= \frac{243 x ^{4}}{4} + \frac{135 x ^{6}}{2} + \frac{135 x ^{8}}{4} + 9 x^{10} + \frac{5 x ^{12}}{4} + \frac{x ^{14}}{14} + C

\int_{1}^{4} x^{2} - 2 x - 15 d x

x \to 0 lim (\frac{6}{x})

2 x lo g_{7} (x)

x \to 3 lim (12)

\int \frac{x ^{2}}{( 64 + x ^{2} ) ^{2}} d x

What is the integral of x^3(3+x^2)^5 ?
The integral of x^3(3+x^2)^5 is (243x^4)/4+(135x^6)/2+(135x^8)/4+9x^{10}+(5x^{12})/4+(x^{14})/(14)+C