Solve the following: Derivatives and anti-derivatives d dx d -In(x): = dx d -arctan(x)= dx d dx

Solve the following: 

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Derivatives and anti-derivatives d dx d -In(x): = dx d -arctan(x)= dx d dx d -sin(x) = cos(x) -cos(x)=-sin(x) -tan(x) = sec²(x) -sec(x)=sec(x) tan(x) -cot(x)=-csc²(x) -csc(x)=-csc(x) cot(x) dx d dx d ● X dx d dx 1+x² |√√x' dx = = = 1 + C fe* dx = e² +C r+1 +C, r*-1 - dx = ln |x|+ C X dx = arctan(x) + C 1+x² [cos(x) dx = sin(x)+C [sin(x) dx = -cos(x) + C [sec²(x) dx =tan(x) + C [sec(x) tan(x) dx = sec(x) + C csc²(x) dx = -cot(x) + C [csc(x) cot(x) dx = -csc(x) + C For example, fe²¹ dx = +¹ +C You may use (and I encourage you to use) what I have been calling the "reverse chain rule factor" to avoid simple u-substitutions. or (₁) + x = (²) + C² - dx = 2 arctan -C and so on. You may not use any other formulas from the table of integrals in your text (although that doesn't mean you can't use those formulas to check your answer). Trig Identities The definitions of tan, cot, sec, csc in terms of sin and cos sin²(x) + cos²(x)=1, tan² (x)+1=sec²(x), 1+cot²(x)=csc²(x) sin(x)=(1-cos(2x)) and cos'(x)= (1+cos(2x)) sin(20) = 2 sin(0) cos(0) and cos(2x) = cos(x)-sin(x) Derivatives and anti-derivatives d dx d -In(x): = dx d -arctan(x)= dx d dx d -sin(x) = cos(x) -cos(x)=-sin(x) -tan(x) = sec²(x) -sec(x)=sec(x) tan(x) -cot(x)=-csc²(x) -csc(x)=-csc(x) cot(x) dx d dx d ● X dx d dx 1+x² |√√x' dx = = = 1 + C fe* dx = e² +C r+1 +C, r*-1 - dx = ln |x|+ C X dx = arctan(x) + C 1+x² [cos(x) dx = sin(x)+C [sin(x) dx = -cos(x) + C [sec²(x) dx =tan(x) + C [sec(x) tan(x) dx = sec(x) + C csc²(x) dx = -cot(x) + C [csc(x) cot(x) dx = -csc(x) + C For example, fe²¹ dx = +¹ +C You may use (and I encourage you to use) what I have been calling the "reverse chain rule factor" to avoid simple u-substitutions. or (₁) + x = (²) + C² - dx = 2 arctan -C and so on. You may not use any other formulas from the table of integrals in your text (although that doesn't mean you can't use those formulas to check your answer). Trig Identities The definitions of tan, cot, sec, csc in terms of sin and cos sin²(x) + cos²(x)=1, tan² (x)+1=sec²(x), 1+cot²(x)=csc²(x) sin(x)=(1-cos(2x)) and cos'(x)= (1+cos(2x)) sin(20) = 2 sin(0) cos(0) and cos(2x) = cos(x)-sin(x)