# How do you integrate #int_0^(pi/2) x^2*sin(2x) dx# using integration by parts?

Now, sine waves can be easily integrated or differentiated again, while polynomials can only be differentiated a finite number of times before becoming zero.

The second integration by parts is coming up now.

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To integrate ( \int_{0}^{\frac{\pi}{2}} x^2 \sin(2x) , dx ) using integration by parts, follow these steps:

- Choose ( u = x^2 ) and ( dv = \sin(2x) , dx ).
- Calculate ( du = 2x , dx ) and ( v = -\frac{1}{2} \cos(2x) ).
- Apply the integration by parts formula: ( \int u , dv = uv - \int v , du ).
- Substitute the values of ( u ), ( v ), ( du ), and ( dv ) into the formula.
- Evaluate the integrals and simplify the expression.

The integration by parts formula yields:

[ \int_{0}^{\frac{\pi}{2}} x^2 \sin(2x) , dx = \left[ -\frac{1}{2}x^2 \cos(2x) \right]*{0}^{\frac{\pi}{2}} + \frac{1}{2} \int*{0}^{\frac{\pi}{2}} \cos(2x) \cdot 2x , dx ]

[ = -\frac{1}{2}\left(\frac{\pi^2}{4}\right) \cos(\pi) - \left( -\frac{1}{2}(0)^2 \cos(0) \right) + \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \cos(2x) \cdot 2x , dx ]

[ = -\frac{\pi^2}{8} + \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \cos(2x) \cdot 2x , dx ]

Now, integrate ( \int_{0}^{\frac{\pi}{2}} \cos(2x) \cdot 2x , dx ) by parts again:

- Choose ( u = 2x ) and ( dv = \cos(2x) , dx ).
- Calculate ( du = 2 , dx ) and ( v = \frac{1}{2} \sin(2x) ).
- Apply the integration by parts formula.
- Evaluate the integrals and simplify the expression.

This yields:

[ \int_{0}^{\frac{\pi}{2}} x^2 \sin(2x) , dx = -\frac{\pi^2}{8} + \left[ \frac{1}{2}x \sin(2x) \right]*{0}^{\frac{\pi}{2}} - \frac{1}{2} \int*{0}^{\frac{\pi}{2}} \sin(2x) , dx ]

[ = -\frac{\pi^2}{8} + \left( \frac{1}{2}\frac{\pi}{2} \sin(\pi) - \frac{1}{2}(0) \sin(0) \right) - \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \sin(2x) , dx ]

[ = -\frac{\pi^2}{8} - \frac{\pi}{4} ]

Thus, ( \int_{0}^{\frac{\pi}{2}} x^2 \sin(2x) , dx = -\frac{\pi^2}{8} - \frac{\pi}{4} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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