What is the integral from 3 to x^4 of sin(5t)*t^5 ?

The integral from 3 to x^4 of sin(5t)*t^5 is

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integral from 3 to x^4 of sin(5t)*t^5

\int_{3}^{x^{4}} sin (5 t) \cdot t^{5} d t

\frac{625 x ^{16} sin ( 5 x ^{4} ) - 300 x ^{8} sin ( 5 x ^{4} ) - 120 x ^{4} cos ( 5 x ^{4} ) + 24 sin ( 5 x ^{4} ) + 500 x ^{12} cos ( 5 x ^{4} ) - 625 x ^{20} cos ( 5 x ^{4} ) + 138735 cos ( 15 ) - 47949 sin ( 15 )}{3125}

Solution steps

\int_{3}^{x^{4}} sin (5 t) t^{5} d t

Apply Integration By Parts

= [\frac{1}{6} t^{6} sin (5 t) - \int \frac{5}{6} t^{6} cos (5 t) d t]_{3}^{x^{4}}

\int \frac{5}{6} t^{6} cos (5 t) d t = \frac{1}{93750} (15625 t^{6} sin (5 t) - 6 (5 (625 t^{4} sin (5 t) - 4 (3 (25 t^{2} sin (5 t) - 2 (- 5 t cos (5 t) + sin (5 t))) - 125 t^{3} cos (5 t))) - 3125 t^{5} cos (5 t)))

= [\frac{1}{6} t^{6} sin (5 t) - \frac{1}{93750} (15625 t^{6} sin (5 t) - 6 (5 (625 t^{4} sin (5 t) - 4 (3 (25 t^{2} sin (5 t) - 2 (- 5 t cos (5 t) + sin (5 t))) - 125 t^{3} cos (5 t))) - 3125 t^{5} cos (5 t)))]_{3}^{x^{4}}

Simplify

[\frac{1}{6} t^{6} sin (5 t) - \frac{1}{93750} (15625 t^{6} sin (5 t) - 6 (5 (625 t^{4} sin (5 t) - 4 (3 (25 t^{2} sin (5 t) - 2 (- 5 t cos (5 t) + sin (5 t))) - 125 t^{3} cos (5 t))) - 3125 t^{5} cos (5 t)))]_{3}^{x^{4}} : [\frac{1}{3125} (625 t^{4} sin (5 t) - 300 t^{2} sin (5 t) - 120 t cos (5 t) + 24 sin (5 t) + 500 t^{3} cos (5 t) - 625 t^{5} cos (5 t))]_{3}^{x^{4}}

= [\frac{1}{3125} (625 t^{4} sin (5 t) - 300 t^{2} sin (5 t) - 120 t cos (5 t) + 24 sin (5 t) + 500 t^{3} cos (5 t) - 625 t^{5} cos (5 t))]_{3}^{x^{4}}

Compute the boundaries:

\frac{1}{3125} (625 x^{16} sin (5 x^{4}) - 300 x^{8} sin (5 x^{4}) - 120 x^{4} cos (5 x^{4}) + 24 sin (5 x^{4}) + 500 x^{12} cos (5 x^{4}) - 625 x^{20} cos (5 x^{4})) - \frac{- 138735 cos ( 15 ) + 47949 sin ( 15 )}{3125}

= \frac{1}{3125} (625 x^{16} sin (5 x^{4}) - 300 x^{8} sin (5 x^{4}) - 120 x^{4} cos (5 x^{4}) + 24 sin (5 x^{4}) + 500 x^{12} cos (5 x^{4}) - 625 x^{20} cos (5 x^{4})) - \frac{- 138735 cos ( 15 ) + 47949 sin ( 15 )}{3125}

Simplify

= \frac{625 x ^{16} sin ( 5 x ^{4} ) - 300 x ^{8} sin ( 5 x ^{4} ) - 120 x ^{4} cos ( 5 x ^{4} ) + 24 sin ( 5 x ^{4} ) + 500 x ^{12} cos ( 5 x ^{4} ) - 625 x ^{20} cos ( 5 x ^{4} ) + 138735 cos ( 15 ) - 47949 sin ( 15 )}{3125}

What is the integral from 3 to x^4 of sin(5t)*t^5 ?
The integral from 3 to x^4 of sin(5t)*t^5 is