Answer to Question #275030 in Calculus for Asif

Question #275030

4. a) Evaluate : integrate (x ^ 3 + 1/(x ^ 3)) dx


b) int 0 ^ pi 2 -4 sin xdx 3+4 cos x

1
Expert's answer
2021-12-08T12:19:03-0500

4.

a)


"\\int (x^3+\\dfrac{1}{x^3})dx=\\dfrac{x^4}{4}-\\dfrac{1}{2x^2}+C"

b)


"\\displaystyle\\int_{0}^{\\pi\/2}\\dfrac{-4\\sin x}{3+4\\cos x}dx"

"\\int \\dfrac{-4\\sin x}{3+4\\cos x}dx"

Use "u" -substitution

"u=3+4\\cos x, du=-4\\sin x"


"\\int \\dfrac{-4\\sin x}{3+4\\cos x}dx=\\int \\dfrac{du}{u}dx=\\ln(|u|)+C"

"=\\ln(|3+4\\cos x|)+C"

"\\displaystyle\\int_{0}^{\\pi\/2}\\dfrac{-4\\sin x}{3+4\\cos x}dx=[\\ln(|3+4\\cos x|)]\\begin{matrix}\n \\pi\/2 \\\\\n 0\n\\end{matrix}"

"=\\ln(|3+4\\cos (\\pi\/2)|)-\\ln(|3+4\\cos (0)|)"

"=\\ln 3-\\ln 7=\\ln (3\/7)"


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