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Popular >

integral of (x^5-x^3+3)^4

Solution

\int (x^{5} - x^{3} + 3)^{4} d x

Solution

\frac{x ^{21}}{21} - \frac{4 x ^{19}}{19} + \frac{6 x ^{17}}{17} + \frac{3 x ^{16}}{4} - \frac{4 x ^{15}}{15} - \frac{18 x ^{14}}{7} + \frac{x ^{13}}{13} + 3 x^{12} + \frac{54 x ^{11}}{11} - \frac{6 x ^{10}}{5} - 12 x^{9} + \frac{54 x ^{7}}{7} + 18 x^{6} - 27 x^{4} + 81 x + C

Solution steps

\int (x^{5} - x^{3} + 3)^{4} d x

Expand

(x^{5} - x^{3} + 3)^{4} : x^{20} - 4 x^{18} + 6 x^{16} + 12 x^{15} - 4 x^{14} - 36 x^{13} + x^{12} + 36 x^{11} + 54 x^{10} - 12 x^{9} - 108 x^{8} + 54 x^{6} + 108 x^{5} - 108 x^{3} + 81

= \int x^{20} - 4 x^{18} + 6 x^{16} + 12 x^{15} - 4 x^{14} - 36 x^{13} + x^{12} + 36 x^{11} + 54 x^{10} - 12 x^{9} - 108 x^{8} + 54 x^{6} + 108 x^{5} - 108 x^{3} + 81 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 x^{20} d x - \int 4 x^{18} d x + \int 6 x^{16} d x + \int 12 x^{15} d x - \int 4 x^{14} d x - \int 36 x^{13} d x + \int x^{12} d x + \int 36 x^{11} d x + \int 54 x^{10} d x - \int 12 x^{9} d x - \int 108 x^{8} d x + \int 54 x^{6} d x + \int 108 x^{5} d x - \int 108 x^{3} d x + \int 81 d x

\int x^{20} d x = \frac{x ^{21}}{21}

\int 4 x^{18} d x = \frac{4 x ^{19}}{19}

\int 6 x^{16} d x = \frac{6 x ^{17}}{17}

\int 12 x^{15} d x = \frac{3 x ^{16}}{4}

\int 4 x^{14} d x = \frac{4 x ^{15}}{15}

\int 36 x^{13} d x = \frac{18 x ^{14}}{7}

\int x^{12} d x = \frac{x ^{13}}{13}

\int 36 x^{11} d x = 3 x^{12}

\int 54 x^{10} d x = \frac{54 x ^{11}}{11}

\int 12 x^{9} d x = \frac{6 x ^{10}}{5}

\int 108 x^{8} d x = 12 x^{9}

\int 54 x^{6} d x = \frac{54 x ^{7}}{7}

\int 108 x^{5} d x = 18 x^{6}

\int 108 x^{3} d x = 27 x^{4}

\int 81 d x = 81 x

= \frac{x ^{21}}{21} - \frac{4 x ^{19}}{19} + \frac{6 x ^{17}}{17} + \frac{3 x ^{16}}{4} - \frac{4 x ^{15}}{15} - \frac{18 x ^{14}}{7} + \frac{x ^{13}}{13} + 3 x^{12} + \frac{54 x ^{11}}{11} - \frac{6 x ^{10}}{5} - 12 x^{9} + \frac{54 x ^{7}}{7} + 18 x^{6} - 27 x^{4} + 81 x

Add a constant to the solution

= \frac{x ^{21}}{21} - \frac{4 x ^{19}}{19} + \frac{6 x ^{17}}{17} + \frac{3 x ^{16}}{4} - \frac{4 x ^{15}}{15} - \frac{18 x ^{14}}{7} + \frac{x ^{13}}{13} + 3 x^{12} + \frac{54 x ^{11}}{11} - \frac{6 x ^{10}}{5} - 12 x^{9} + \frac{54 x ^{7}}{7} + 18 x^{6} - 27 x^{4} + 81 x + C

Popular Examples

integral of x^2(3-10x^3)^4

\int x^{2} (3 - 10 x^{3})^{4} d x

integral of-2/(x(ln(x))^3)

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

integral of (e^x)/((9-e^x)^2)

\int \frac{e ^{x}}{( 9 - e ^{x} ) ^{2}} d x

integral of-4/(x(ln(x))^4)

\int - \frac{4}{x ( ln ( x ) ) ^{4}} d x

integral of 1/((x^2+1)(x+1)^2)

\int \frac{1}{( x ^{2} + 1 ) ( x + 1 ) ^{2}} d x

Frequently Asked Questions (FAQ)

What is the integral of (x^5-x^3+3)^4 ?
The integral of (x^5-x^3+3)^4 is