Week 6 Integration 1 Flashcards
how is the area under a curve found
integration
what is the integration addition rule
∫f(x) + g(x) dx = ∫f(x)dx + ∫g(x)dx
what is the integration constant rule
∫cf(x)dx = c∫f(x)dx
what is the integration reversed limits rule
b a
∫f(x) dx = - ∫f(x) dx
a b
what is the splitting up the range integration rule
b c b
∫f(x) dx = ∫f(x)dx + ∫f(x)dx
a a c
what is unique of odd functions integrated over symmetric limits (a, -a)
the integral is equal to 0
what is unique of even function integrated over symmetric limits (a, -a)
it is equal to two times the integration of one of the limits and 0 so the limits on this new integral are a and 0
what is the value of the perfect integral
∫ f’(x) / f(x) dx
ln[f(x)] + c
what is the value of the perfect integral
∫ f(x)f’(x) dx
1/2 f^2(x) +c
what is the value of the perfect integral
∫ f’(x) / sqrt[f(x)] dx
2[f(x)]^1/2 + c
how do we determine if an integral with a singularity diverges or not
we integrate close to the singularity
what is the 4 step procedure for integration by substiution
1 choose a reasonable substitution
2 differentiate to change variable of the integration
3 make the substitution
4 perform the integration
what is the value of the following standard integral and the appropriate substitution
∫ 1 / sqrt[a^2 - x^2]
value is arcsin(x/a) + c
substitution is x = asin(u)
what is the value of the following standard integral and the appropriate substitution
∫ 1 / sqrt[x^2 - a^2]
value is arccosh(x/a) + c
substitution is x = acosh(u)
what is the value of the following standard integral and the appropriate substitution
∫ 1 / sqrt[x^2 + a^2]
value is arcsinh(x/a) + c
substitution is x = asinh(u)
