How do you integrate #intx^3sqrt(16 - x^2) dx#?
Normally here would be okay, but we can go a bit further.
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To integrate ( \int x^3 \sqrt{16 - x^2} , dx ), you can use the substitution method. Let ( u = 16 - x^2 ). Then ( du = -2x , dx ). This implies ( x , dx = -\frac{1}{2} du ).
Substituting these into the integral, you get:
[ \int x^3 \sqrt{16 - x^2} , dx = \int -\frac{1}{2} \left(16 - u\right)^{\frac{1}{2}} \cdot u , du ]
Now, you can use integration by parts. Let ( dv = u , du ) and ( v = \frac{2}{3} (16 - u)^{\frac{3}{2}} ).
[ \int -\frac{1}{2} \left(16 - u\right)^{\frac{1}{2}} \cdot u , du = -\frac{1}{2} \cdot \frac{2}{3} (16 - u)^{\frac{3}{2}} \cdot u + \frac{1}{2} \int \frac{2}{3} (16 - u)^{\frac{3}{2}} , du ]
Now, integrate the remaining integral:
[ \frac{1}{2} \int \frac{2}{3} (16 - u)^{\frac{3}{2}} , du = -\frac{1}{3} (16 - u)^{\frac{3}{2}} + C ]
Combine the results:
[ \int x^3 \sqrt{16 - x^2} , dx = -\frac{1}{3} (16 - u)^{\frac{3}{2}} - \frac{1}{2} \cdot \frac{2}{3} (16 - u)^{\frac{3}{2}} \cdot u + C ]
Replace ( u ) with ( 16 - x^2 ) to get the final result:
[ \int x^3 \sqrt{16 - x^2} , dx = -\frac{1}{3} (16 - 16 + x^2)^{\frac{3}{2}} - \frac{1}{3} (16 - x^2)^{\frac{3}{2}} + C ]
[ \int x^3 \sqrt{16 - x^2} , dx = -\frac{1}{3} (x^2)^{\frac{3}{2}} - \frac{1}{3} (16 - x^2)^{\frac{3}{2}} + C ]
[ \int x^3 \sqrt{16 - x^2} , dx = -\frac{1}{3} x^3 - \frac{1}{3} (16 - x^2)^{\frac{3}{2}} + C ]
This is the final result after integration.
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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.
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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