Anti-derivatives of sums and scalar multiples, using linearity together with standard forms such as ∫xndx=n+1xn+1+C,
The integrals of sin and cos are
The anti-derivative of a sum is the sum of the anti-derivatives:
The anti-derivative of a scalar multiple (a constant that does not depend on the integrating variable) can pass through the integral:
x1 is defined for x<0, but lnx is not. Specifically:
So x1 is the derivative of lnx and of ln(−x).
However, we can simplify further. Recall the definition of the absolute value:
Hence, we have
Nice work completing Anti-Derivative Rules, here's a quick recap of what we covered:
Exercises checked off
Anti-derivatives of sums and scalar multiples, using linearity together with standard forms such as ∫xndx=n+1xn+1+C,
The integrals of sin and cos are
The anti-derivative of a sum is the sum of the anti-derivatives:
The anti-derivative of a scalar multiple (a constant that does not depend on the integrating variable) can pass through the integral:
x1 is defined for x<0, but lnx is not. Specifically:
So x1 is the derivative of lnx and of ln(−x).
However, we can simplify further. Recall the definition of the absolute value:
Hence, we have
Nice work completing Anti-Derivative Rules, here's a quick recap of what we covered:
Exercises checked off