integral from 0 to 3 of x^2cos((npix)/3)

الحلّ

\int_{0}^{3} x^{2} cos (\frac{nπ x}{3}) d x

\frac{54 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

خطوات الحلّ

\int_{0}^{3} x^{2} cos (\frac{nπ x}{3}) d x

فعّل التكامل بالتجزيء

= [\frac{3}{πn} x^{2} sin (\frac{πn x}{3}) - \int \frac{6}{πn} x sin (\frac{πn x}{3}) d x]_{0}^{3}

\int \frac{6}{πn} x sin (\frac{πn x}{3}) d x = \frac{6}{π ^{3} n ^{3}} (- 3 πn x cos (\frac{πn x}{3}) + 9 sin (\frac{πn x}{3}))

= [\frac{3}{πn} x^{2} sin (\frac{πn x}{3}) - \frac{6}{π ^{3} n ^{3}} (- 3 πn x cos (\frac{πn x}{3}) + 9 sin (\frac{πn x}{3}))]_{0}^{3}

احسب التكامل المحدّد:

\frac{54 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

= \frac{54 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

\int_{0}^{2} 16 x - x^{5} d x

\int_{1}^{2} \frac{3}{7} x^{2} x^{2} d x

\int_{2}^{4} 3 + x^{2} d x

\int_{2}^{6} 6 x^{2} ln (3 x) d x

\int_{0}^{k} sin^{2} (10 π t) d t