QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
7. Evaluate $\int 3 x^{3} \cos \left(x^{4}\right) d x$
8. Evaluate $\int_{7}^{- 7}- 4 x^{5}- 6 x^{3}+ 54 x d x$
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Answer
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Step 1
\int \frac{3}{4} \cos(u) u^{-3/4} du
To evaluate the integral in part (a), we can use integration by substitution. Substituting these into the integral, we get: Therefore, our integral becomes:
Step 2
\int \frac{3}{4} \cos(u) u^{-3/4} du = \frac{3}{4} \left[ \sin(u) u^{-3/4} + \int \frac{3}{4} \sin(u) u^{-7/4} du \right]
To solve this integral, we can use integration by parts.
Final Answer
\int 3 x^{3} \cos \left(x^{4}\right) d x = \frac{3}{4} \sin \left(x^{4}\right) x^{- 3} + C For part (b), we can use the power rule for integration: \int_{7}^{- 7}- 4 x^{5}- 6 x^{3}+ 54 x d x = \left. - 4 \frac{x^{6}}{6} - 6 \frac{x^{4}}{4} + 54 \frac{x^{2}}{2} \right]_{- 7}^{7} 5. Evaluating the antiderivative at the limits of integration, we get: \int_{7}^{- 7}- 4 x^{5}- 6 x^{3}+ 54 x d x = -\frac{4}{6} (7^{6} - (- 7)^{6}) - \frac{6}{4} (7^{4} - (- 7)^{4}) + \frac{54}{2} (7^{2} - (- 7)^{2}) 6. Simplifying, we have: \int_{7}^{- 7}- 4 x^{5}- 6 x^{3}+ 54 x d x = -\frac{4}{6} (1176) - \frac{6}{4} (2401) + \frac{54}{2} (98) \int_{7}^{- 7}- 4 x^{5}- 6 x^{3}+ 54 x d x = - 84 - 360.125 + 2646 7. \int_{7}^{- 7}- 4 x^{5}- 6 x^{3}+ 54 x d x = 2262 - 444 = \boxed{1818}
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