For positive integers n ≥ 1, consider the functions 2 A (a) Show that (b) Show that Sn(x) = ㅠ [ Sn = sin(nx) [" (5(2))² de 1 for all n. da dx = Sn(x) Sm(x) dx = 0 for all n m.

Q: 12 If [" f(x) dx = 19 and ' [₁ f(x) dx [R²F(x) f(x) dx = 4, find the following.

A: We can answer the question as below using the properties of definite integral.

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A: To evaluate the integral

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Q: 7. Prove that d dx [cos ¹ x] = = 1 √1-x² x2

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A: We have to calculate

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A:

Q: 3 a) Given fg(x)dx=5, [[3g(x)– 4h(x)]dx = 27 . i) Find [h(x)dx . ii) Hence, evaluate [4x – 3h(x)]dx.

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Q: b Find 60 such that - So (x − 1)dx = b == = 0.

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Q: Calculate M6 for f(x) = 5 · In(æ³) over [1, 2].

A: Given:   f(x)=5.ln(x3) over [1,3]

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A: Bessel functions are a family of mathematical functions that arise in many areas of physics and…