To calculate the integral / sin (x) e COS( ) dac, we must first use the substitution method with u = to get the full / F( u ) d where F(u) = To evaluate the integral / F( u ) d we use the method of integration by parts with f(u) = et g (u) = to get (F(u) du = G(u) - /H(u) du, where G(u) = et H(u) = Then, to calculate the integral / H( u ) d we must use the method of integration by parts a second time with f (u) = et g (u) = to get SH(u ) du = +C (Do not add the integration constantC'because we have already done this for you.) We then find that [F(u) du = +C and so sing (a) ecos() de = +CWe want to calculate the integral (4x + 3 x + 5 ) edking two (and only two) integrations by parts. This means that you have to make a good choice off et g for the two integrations by parts. For the first integration by parts, we have to choose [f(x),g' (x)] = Attention : For this question, you must use the notation of scientific calculators and you must not forget the brackets (square parentheses); in particular, the symbol * must be used for all multiplications (eg2 ais written 2*x .) to get ( 42 + 32 + 5) ex du = G(x) - /H(x)da, where G(@) = et H(x) = Then, to integrate / H(x) da, we must use the method of integration by parts a second time with [f(z),g (x)] = Attention : For this question, you must use the notation of scientific calculators and you must not forget the brackets (square parentheses); in particular, the symbol * must be used for all multiplications (eg2 ais written 2*x .) to get H(x) do = +C (do not add the integration constantC'because we did it for you.) We have therefore found that ( 42 2 + 3 x + 5 ex da = +CCalculate the indefinite integral 6x In (x +2) dx = + C, whereC represents the constant of integration. So don't add an integration constant in your answer since we did it for you. Here, we are looking for a valid primitive fora + 2 > 0. Do not use absolute values inside logarithms.We want to use the method of integration by parts to evaluate the indenite integral 11:1 3 f ( ) dz 34 f 11:)[132 ff{z)gf(2:)dm What are the best choices for f et grin the formula flm) = _. r g (m) = It follows that l' f (m) = _. 9(3) = 80. applying the method of integration by parts. we conclude that 111(2) do: = +0 2:4 whereCrepresents the constant of integration. So don't add an integration constant in your answer since we did it for you. Here. we are looking for a valid primitive fora: > I]. Do not use absolute values inside logarithms. Using a variety of methods, determine which of the following formulas represents / e" ( cos (x) + sin (x) ) dx . (a) er cos (x) + el sin (x) + C (b)et ( sin (x) - cos (x)) + C (c) et cos (x) + C (d) e sin (z) + C