\int_{0}^{4}x\left(4-x\right)\sin\left(\frac{n\pi x}{4}\right)dx

asked by guest
on Nov 16, 2024 at 10:32 pm



You asked:

Evaluate the integral: \(\int_{0}^{4}{x \cdot \left(4 - x\right) \cdot \sin\left( \frac{n \cdot \pi \cdot x}{4} \right)}dx\)

MathBot Answer:

Evaluated



\(\displaystyle \int_{0}^{4}{x \cdot \left(4 - x\right) \cdot \sin\left( \frac{n \cdot \pi \cdot x}{4} \right)}dx = \begin{cases} - \frac{64 \sin{\left(n \pi \right)}}{n^{2} \pi^{2}} - \frac{128 \cos{\left(n \pi \right)}}{n^{3} \pi^{3}} + \frac{128}{n^{3} \pi^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} \)

Expanded

\[\int_{0}^{4}{x \cdot \left(4 - x\right) \cdot \sin\left( \frac{n \cdot \pi \cdot x}{4} \right)}dx = \int\limits_{0}^{4} \left(- x^{2} \sin{\left(\frac{n x \pi}{4} \right)}\right)\, dx + \int\limits_{0}^{4} 4 x \sin{\left(\frac{n x \pi}{4} \right)}\, dx\]

Factored

\[\int_{0}^{4}{x \cdot \left(4 - x\right) \cdot \sin\left( \frac{n \cdot \pi \cdot x}{4} \right)}dx = - \int\limits_{0}^{4} x \left(x - 4\right) \sin{\left(\frac{n x \pi}{4} \right)}\, dx\]

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