3. The following is an incorrect solution to evaluating the integral of dx/(1 + 4x^2)^3/2 . Set x = tan θ. Then the integral becomes: the integral of dx/(1 + 4x^2)^3/2 = integral of dθ^(1 + 4 tan2 θ)^3/2 = intgral of dθ/ (sec^2 θ)^3/2 =integral of dθ/sec^3 θ =intgral of cos^3θ dθ = integral of (1 − sin^2 θ) cos θ dθ =integral of (1 − u^2) du = u − u^3/3+ C = sin θ − sin^3 θ/3 + C *Use u = sin θ, du = cos θ dθ. (a) There are MULTIPLE errors in the above work. Identify them all. (b) Provide a correct solution to evaluating this integral.
3. The following is an incorrect solution to evaluating the integral of dx/(1 + 4x^2)^3/2 . Set x = tan θ. Then the integral becomes: the integral of dx/(1 + 4x^2)^3/2 = integral of dθ^(1 + 4 tan2 θ)^3/2 = intgral of dθ/ (sec^2 θ)^3/2 =integral of dθ/sec^3 θ =intgral of cos^3θ dθ = integral of (1 − sin^2 θ) cos θ dθ =integral of (1 − u^2) du = u − u^3/3+ C = sin θ − sin^3 θ/3 + C *Use u = sin θ, du = cos θ dθ. (a) There are MULTIPLE errors in the above work. Identify them all. (b) Provide a correct solution to evaluating this integral.
3. The following is an incorrect solution to evaluating the integral of dx/(1 + 4x^2)^3/2 . Set x = tan θ. Then the integral becomes: the integral of dx/(1 + 4x^2)^3/2 = integral of dθ^(1 + 4 tan2 θ)^3/2 = intgral of dθ/ (sec^2 θ)^3/2 =integral of dθ/sec^3 θ =intgral of cos^3θ dθ = integral of (1 − sin^2 θ) cos θ dθ =integral of (1 − u^2) du = u − u^3/3+ C = sin θ − sin^3 θ/3 + C *Use u = sin θ, du = cos θ dθ. (a) There are MULTIPLE errors in the above work. Identify them all. (b) Provide a correct solution to evaluating this integral.
3. The following is an incorrect solution to evaluating the integral of dx/(1 + 4x^2)^3/2 . Set x = tan θ. Then the integral becomes: the integral of dx/(1 + 4x^2)^3/2 = integral of dθ^(1 + 4 tan2 θ)^3/2
= intgral of dθ/ (sec^2 θ)^3/2
=integral of dθ/sec^3 θ
=intgral of cos^3θ dθ
= integral of (1 − sin^2 θ) cos θ dθ
=integral of (1 − u^2) du
= u − u^3/3+ C
= sin θ − sin^3 θ/3 + C *Use u = sin θ, du = cos θ dθ.
(a) There are MULTIPLE errors in the above work. Identify them all.
(b) Provide a correct solution to evaluating this integral.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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