Consider the following integral: This may be evaluated by using integration by parts. -) The product 6z cos(z) must be written in the form u du Determine u, and u -) Apply the integration by parts formula: ) Hence write down the indefinite integral: U= V= fudr = fu#dz = 6x cos(z) dr uv Hete [6x cos(z) dz (Don't forget the constant of integration as this is an indefinite integral.) dz dz

Q: Use the substitution formula to evaluate the integral. 0 (1 + cos (5t)) sin (5t)dt

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Q: Find the following integral: i) Sdx S dx 3x2+1 ii) Sx/(x+1) dx

A: We have to find the integral  ∫x3x2+1 dx ∫xx+1 dx

Q: Rewrite the integration in terms of u. 2x dx = du x² + 16

A: Given :∫032xx2+16dx

Q: 16. Determine the following integrals: (a)Scos 8.xdx. S sin" x.cos.xdx (b)? sec² xdx (c) (d)Se*dx

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Q: u = 1 + We can now make the substitution (9x)4 and evaluate the definite integral with respect to u.…

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Q: 2) Evaluate each integral below: a) cos(x²) x dx c) (2x-1)2 dx

A: In this question we will solve both indefinite integral.

Q: 1. Fill in the blank with the appropriate integration rule (You do not need to name it): Je"dr =…

A: To find the following

Q: Q3) Evaluate the following integral by using Trigonometric substitutions Method. 1 dx Vx2 – 9 y

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Q: Integrate sinx given integral by 20 2 w5 x= sing the using sin: 1 + si

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Q: Q3) Use substitution to find the following integrals a. f sec²x √tan x dx b. dx x3 Sa (1+x4)5 C. csc…

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Q: 1- x dx 21 y =

A: We solve the given integral using Trigonometry substitutions method.

Q: a) The value of the following integral 224 2da is Answer: I b) Find the indefinite integral (2x+3)dr…

A: To evaluate the following integral.a ∫012x4-2 dxb ∫2x+3 dxc ∫0x et dt

Q: Evaluate one the integrals: B E b) cos(8x) x² +1 0 -dx

A: 1.If f(x)=p(x)q(x) and q(x) have only complex roots then ∫-∞∞f(x)dx=2πi[Sum of Residues of f(z) at…