Integral of Polynomials

\int (4 x^{7} + 8 x^{6} + 4 x^{5} + 8 x^{4} + 6 x^{3} + 3 x^{2} - 5 x + 8) d x

=

\int 4 x^{7} d x + \int 8 x^{6} d x + \int 4 x^{5} d x + \int 8 x^{4} d x + \int 6 x^{3} d x + \int 3 x^{2} d x + \int -5 x d x + \int 8 d x

=

4 \int x^{7} d x + 8 \int x^{6} d x + 4 \int x^{5} d x + 8 \int x^{4} d x + 6 \int x^{3} d x + 3 \int x^{2} d x - 5 \int x d x + 8 \int 1 d x

=

4 (\frac{1}{8} x^{8}) + 8 (\frac{1}{7} x^{7}) + 4 (\frac{1}{6} x^{6}) + 8 (\frac{1}{5} x^{5}) + 6 (\frac{1}{4} x^{4}) + 3 (\frac{1}{3} x^{3}) - 5 (\frac{1}{2} x^{2}) + 8 (x) + C

=

\frac{1}{2} x^{8} + \frac{8}{7} x^{7} + \frac{2}{3} x^{6} + \frac{8}{5} x^{5} + \frac{3}{2} x^{4} + x^{3} - \frac{5}{2} x^{2} + 8 x + C

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Copyright © Dayal D. Purohit, Ph.D.(Mathematics)