How do you integrate #intx^(2/3) *ln x# from 1 to 4?
With these choices, we get:
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To integrate the function ( \int x^{\frac{2}{3}} \ln(x) ) from 1 to 4, use integration by parts.
Let ( u = \ln(x) ) and ( dv = x^{\frac{2}{3}} dx ). Then, ( du = \frac{1}{x} dx ) and ( v = \frac{3}{5}x^{\frac{5}{3}} ).
Now apply the integration by parts formula:
[ \int u , dv = uv - \int v , du ]
Substitute the values:
[ \int x^{\frac{2}{3}} \ln(x) , dx = \frac{3}{5}x^{\frac{5}{3}} \ln(x) - \int \frac{3}{5}x^{\frac{5}{3}} \cdot \frac{1}{x} , dx ]
Simplify the integral:
[ = \frac{3}{5}x^{\frac{5}{3}} \ln(x) - \frac{3}{5} \int x^{\frac{2}{3}} , dx ]
Now integrate the remaining integral:
[ = \frac{3}{5}x^{\frac{5}{3}} \ln(x) - \frac{3}{5} \left( \frac{3}{5}x^{\frac{5}{3}} \right) + C ]
Evaluate the integral from 1 to 4:
[ = \left[ \frac{3}{5}x^{\frac{5}{3}} \ln(x) - \frac{9}{25}x^{\frac{5}{3}} \right]_1^4 ]
[ = \left( \frac{3}{5} \cdot 4^{\frac{5}{3}} \ln(4) - \frac{9}{25} \cdot 4^{\frac{5}{3}} \right) - \left( \frac{3}{5} \cdot 1^{\frac{5}{3}} \ln(1) - \frac{9}{25} \cdot 1^{\frac{5}{3}} \right) ]
[ = \left( \frac{3}{5} \cdot 4^{\frac{5}{3}} \ln(4) - \frac{9}{25} \cdot 4^{\frac{5}{3}} \right) - \left( 0 - \frac{9}{25} \right) ]
[ = \frac{3}{5} \cdot 4^{\frac{5}{3}} \ln(4) - \frac{9}{25} \cdot 4^{\frac{5}{3}} + \frac{9}{25} ]
[ ≈ 9.0407 ]
So, the value of the integral of ( \int_{1}^{4} x^{\frac{2}{3}} \ln(x) , dx ) is approximately 9.0407.
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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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