72. (a) Use integration by parts to show that, for any positive integer n, dx (x² + a²)" dx 2a (n - 1)(x² + a²)«-1 2n - 3 dx 2a (n – 1) (x² + a²)«-T (b) Use part (a) to evaluate dx dx and (x² + 1)² (x² + 1)

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A: We have∫x23+4x−4x232dx

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A: Using method of substitution

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A: SOLUTION-

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A: The solution are next step ..

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A: To find the indefinite integral: ∫4x3-9xxdx Solution: Evaluating the given integral.…

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Q: 15. Indefinite Integrals (Part 1) а) -+ cscxcot x dx 2 cosx--sinx+4csc². 3 b) 1 + sec² x dx 1+x2 c)

A: topic - indefinite integral

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Q: 9. S sec³ x tan x dx

A: Given ∫sec3xtanxdx We know that  ∫secx.tanx=secx + c ∫xndx=xn+1n+1+c

Q: ∫cosn(x)dx=1ncosn−1(x)sin(x)+n−1n∫cosn−2(x)dx∫cosn⁡(x)dx=1ncosn−1⁡(x)sin⁡(x)+n−1n∫cosn−2⁡(x)dx to…

A: To find the integration

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A: The given integral is

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A: use substitution method for finding  integral

Q: By using integration by parts, one can prove the reduction formula. | (In(x))" dx = x(In(x))" – n /…

A: This question is about application of integration

Q: 1. Find the numerical Integration of cos(x) + sin(x) where n = 4 using trapezoidal rule.

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