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Evaluate the following : int(1)^(3)x^(...

Evaluate the following :
`int_(1)^(3)x^(2)log xdx.`

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AI Generated Solution

To evaluate the integral \( \int_{1}^{3} x^{2} \log x \, dx \), we will use integration by parts. The formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \] ### Step 1: Choose \( u \) and \( dv \) Let: ...
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NAVNEET PUBLICATION - MAHARASHTRA BOARD-DEFINITE INTEGRALS-Examples for Practice
  1. Evaluate the following definite integral: int0^(pi//2)x^2cosx\ dx

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  2. Evaluate the following : int(0)^(pi//4)x sec^(2)xdx

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  3. Evaluate the following : int(1)^(3)x^(2)log xdx.

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  4. Evaluate the following : int(0)^(2)(1)/(4+x-x^(2))dx

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  5. Evaluate the following : int(-2)^(1)(dx)/(x^(2)+4x+13)

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  6. Evaluate the following definite integral: int0^4 1/(sqrt(4x-x^2))dx

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  7. Evaluate the following : int(0)^(4)(1)/(sqrt(x^(2)+2x+3)dx

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  8. Evaluate the following : int(2)^(5)(1)/(sqrt(x+4)+sqrt(x-2))dx.

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  9. If int1^a (3x^2 + 2x +1)dx = 11 then the value of a is

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  10. Evaluate the following : int(0)^(1)x^(2)sqrt(x^(3)+1)dx

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  11. Evaluate the following : int(0)^(1)x.sqrt((1-x^(2))/(1+x^(2)))dx

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  12. Evaluate the following integrals: int0^1sqrt((1-x)/(1+x))dx

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  13. Evaluate the following : int(0)^(pi//2)(sin^(2)x)/((1+cosx)^(2))dx

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  14. int(0)^(pi//4)(tan^(3)x)/((1+cos2x))dx

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  15. int0^(pi/4) tan^4xdx=

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  16. Evaluate the following : int(0)^(pi//4)sec^(4)xdx.

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  17. Evaluate the following : int(0)^(pi//6)(dx)/(3-2 sin^(2)x)

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  18. Evaluate the following : int(0)^(pi//4)(dx)/(3+4 cos^(2)x)dx

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  19. Evaluate the following : int(0)^(pi//4)(dx)/(3 cos 2x+5)

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  20. Evaluate the following : int(0)^(pi//2)(1)/(5+4 cos x)dx

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