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Evaluate the following definite integral...

Evaluate the following definite integrals: `\int_{(-pi)}^(pi) frac{x(1+sin x)}{1+ cos^2x}` dx

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CHETANA PUBLICATION-Definite integration-Exercise
  1. Evaluate the following definite integrals: \int{0}^(pi) x sin^2 x dx

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  2. Evaluate the following definite integrals: \int{0}^(pi) cos^2 x dx

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  3. Evaluate the following definite integrals: \int{(-pi)}^(pi) frac{x(1+...

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  4. Evaluate the following definite integrals: \int{0}^3 x[x]dx, [x] is g...

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  5. Evaluate the following definite integrals: \int{0}^1 x^2 sqrt (1-x) dx

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  6. Evaluate the following definite integrals: \int{0}^((pi)/2) frac {root...

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  7. Evaluate the following definite integrals: \int{0}^((pi)/2) log(sin x)...

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  8. Evaluate the following definite integrals: \int{(-pi)/2}^((pi)/2) sin^...

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  9. Evaluate the following definite integrals: \int{(-pi)/2}^((pi)/2) log(...

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  10. Prove that \int{a}^a f(x) dx = 0

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  11. Prove that \int{a}^b f(x) dx = -\int{b}^a f(x) dx

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  12. Prove that \int{a}^b f(x) dx = \int{a}^b f(t) dt

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  13. Prove that \int{a}^b f(x) dx = \int{a}^c f(x) dx + \int{c}^b f(x) dx, ...

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  14. Prove that \int{a}^b f(x) dx = \int{a}^b f(a+b-x) dx

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  15. Prove that \int{0}^a f(x) dx = \int{0}^a f(a-x) dx

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  16. Prove that \int{0}^(2a) f(x) dx = \int{0}^a [f(x) + f(2a-x)] dx

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  17. Prove that \int{-a}^a f(x) dx = 0, if f(x) is odd function, =2\int{0}...

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  18. Evaluate the following using limit of a sum: \int{0}^a (3x-1) dx

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  19. Evaluate the following using limit of a sum: \int{2}^5 (3x-2x^2) dx

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  20. Evaluate the following using limit of a sum: \int{0}^4 (1+e^x) dx

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