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Popular >

integral of x^4sqrt(x+4)

Solution

\int x^{4} x + 4 d x

Solution

\frac{2}{11} (x + 4)^{\frac{11}{2}} - \frac{32}{9} (x + 4)^{\frac{9}{2}} + \frac{192}{7} (x + 4)^{\frac{7}{2}} - \frac{512}{5} (x + 4)^{\frac{5}{2}} + \frac{512}{3} (x + 4)^{\frac{3}{2}} + C

Solution steps

\int x^{4} x + 4 d x

Apply u-substitution

: \int (u - 4)^{4} u d u

= \int (u - 4)^{4} u d u

Expand

(u - 4)^{4} u : u^{\frac{9}{2}} - 16 u^{\frac{7}{2}} + 96 u^{\frac{5}{2}} - 256 u^{\frac{3}{2}} + 256 u

= \int u^{\frac{9}{2}} - 16 u^{\frac{7}{2}} + 96 u^{\frac{5}{2}} - 256 u^{\frac{3}{2}} + 256 u d u

Apply the Sum Rule:

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

= \int u^{\frac{9}{2}} d u - \int 16 u^{\frac{7}{2}} d u + \int 96 u^{\frac{5}{2}} d u - \int 256 u^{\frac{3}{2}} d u + \int 256 u d u

\int u^{\frac{9}{2}} d u = \frac{2}{11} u^{\frac{11}{2}}

\int 16 u^{\frac{7}{2}} d u = \frac{32}{9} u^{\frac{9}{2}}

\int 96 u^{\frac{5}{2}} d u = \frac{192}{7} u^{\frac{7}{2}}

\int 256 u^{\frac{3}{2}} d u = \frac{512}{5} u^{\frac{5}{2}}

\int 256 u d u = \frac{512}{3} u^{\frac{3}{2}}

= \frac{2}{11} u^{\frac{11}{2}} - \frac{32}{9} u^{\frac{9}{2}} + \frac{192}{7} u^{\frac{7}{2}} - \frac{512}{5} u^{\frac{5}{2}} + \frac{512}{3} u^{\frac{3}{2}}

Substitute back

u = x + 4

= \frac{2}{11} (x + 4)^{\frac{11}{2}} - \frac{32}{9} (x + 4)^{\frac{9}{2}} + \frac{192}{7} (x + 4)^{\frac{7}{2}} - \frac{512}{5} (x + 4)^{\frac{5}{2}} + \frac{512}{3} (x + 4)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{11} (x + 4)^{\frac{11}{2}} - \frac{32}{9} (x + 4)^{\frac{9}{2}} + \frac{192}{7} (x + 4)^{\frac{7}{2}} - \frac{512}{5} (x + 4)^{\frac{5}{2}} + \frac{512}{3} (x + 4)^{\frac{3}{2}} + C

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Frequently Asked Questions (FAQ)

What is the integral of x^4sqrt(x+4) ?
The integral of x^4sqrt(x+4) is