how to do simple integration?

Question

for the integral cos^3 x dx...i broke up cos^3 into (1-sin^2(x)) (cos(x)) dx                    then through u substitution my u was sin x and my du was cosx dx         so then my final integral is    the integral of (1-u^2) du.    I know the integral of that is -du (u^2-1)x     but only through wolfram integral website...My teacher always gets to the final point and says, oh that's a simple enough integral, and it's been so long i forgot how to do that final integration...Could you please explain   thank you

intc_escapee · Accepted Answer

∫ cos³(x) dx
= 1/4 ∫ 3 cos(x) + cos(3x) dx
= 3/4 sin(x) + 1/12 sin(3x) + C

Answer: 1/12 (9 sin(x) + sin(3x)) + C

? · Answer

5 * dx / (25 + x^2) enable x = 5 * tan(t) and dx might then be 5 * sec(t)^2 * dt 5 * 5 * sec(t)^2 * dt / (25 + 25 * tan(t)^2) => 25 * sec(t)^2 * dt / (25 * sec(t)^2) => dt combine t + C resolve for t x = 5 * tan(t) x/5 = tan(t) arctan(x/5) = t arctan(x/5) + C is the respond.

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how to do simple integration?

2 Answers