evaluate the following general antiderivatives b 7cosx5sinx3

Solution

integral (3 e^x-5/x+7/(1+x^2)+7 cos(x)+7 sec^2(x)+5 sin(x)) dx Integrate the sum term by term and factor out constants: = 7 integral 1/(x^2+1) dx+3 integral e^x dx-5 integral 1/x dx+5 integral sin(x) dx+7 integral cos(x) dx+7 integral sec^2(x) dx The integral of 1/(x^2+1) is tan^(-1)(x): = 7 tan^(-1)(x)+3 integral e^x dx-5 integral 1/x dx+5 integral sin(x) dx+7 integral cos(x) dx+7 integral sec^2(x) dx The integral of sec^2(x) is tan(x): = 7 tan^(-1)(x)+7 tan(x)+3 integral e^x dx-5 integral 1/x dx+5 integral sin(x) dx+7 integral cos(x) dx The integral of e^x is e^x: = 3 e^x+7 tan^(-1)(x)+7 tan(x)-5 integral 1/x dx+5 integral sin(x) dx+7 integral cos(x) dx The integral of 1/x is log(x): = 3 e^x-5 log(x)+7 tan^(-1)(x)+7 tan(x)+5 integral sin(x) dx+7 integral cos(x) dx The integral of cos(x) is sin(x): = 3 e^x-5 log(x)+7 sin(x)+7 tan^(-1)(x)+7 tan(x)+5 integral sin(x) dx The integral of sin(x) is -cos(x): = 3 e^x-5 log(x)+7 sin(x)-5 cos(x)+7 tan^(-1)(x)+7 tan(x)+constant