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

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on Nov 16, 2024 at 10:32 pm



You asked:

Evaluate the integral: ∫04x(4−x)⋅sin⁡(nπx4)dx\int_{0}^{4}{x \left(4 - x\right) \cdot \sin\left( \frac{n \pi x}{4} \right)}dx

MathBot Answer:

Evaluated



∫04x(4−x)⋅sin⁡(nπx4)dx={−64sin⁡(nπ)n2π2−128cos⁡(nπ)n3π3+128n3π3for n>−∞∧n<∞∧n≠00otherwise\displaystyle \int_{0}^{4}{x \left(4 - x\right) \cdot \sin\left( \frac{n \pi 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

∫04x(4−x)⋅sin⁡(nπx4)dx=∫04(−x2sin⁡(nxπ4)) dx+∫044xsin⁡(nxπ4) dx\int_{0}^{4}{x \left(4 - x\right) \cdot \sin\left( \frac{n \pi 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

∫04x(4−x)⋅sin⁡(nπx4)dx=−∫04x(x−4)sin⁡(nxπ4) dx\int_{0}^{4}{x \left(4 - x\right) \cdot \sin\left( \frac{n \pi 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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