Definitions
In the definition of the indefinite integral,
∫ f(x) dx ＝ F(x) + C
what is the meaning of the C?
C is an arbitrary consant. (The constant of integration).
Find the exact value of
int _{-2}^2 |x|dx
int _{-2}^2 |x|dx =4
int (1-1/w)cos(w-ln(w))dw
Rewrite the integral using u-sub!
int (1-1/w)cos(w-ln(w))dw = int cos(u) du
int dx
int dx = x+C
Evaluate
\int_2^6 (3x^2+2x+1)dx
244
In the definition of the indefinite integral,
∫ f(x) dx ＝ F(x) + C
what is the meaning of the F(x) ?
F(x) is an antiderivative of f(x) . That is, F'(x)=f(x).
Estimate the value of int_1^2 x^2 dx using 3 equal width, right-end point rectangles. Do not simplify.
int_1^2 x^2 dx\approx (1/3)f(4/3)+(1/3)f(5/3)+(1/3)f(2)
=(1/3)(4/3)^2+(1/3)(5/3)^2+(1/3)(2)^2
int cos(5x)dx
Rewrite the integral using u-sub!
int cos(5x)dx = 1/5 int cos(u) du
\int x^n dx
when n\ne -1
\int x^n dx = x^{n+1}/{n+1}+C
Evaluate
\int_-3^0 d/dt [t^2(1-t^2)]dt
\int_-3^0 d/dt [t^2(1-t^2)]dt = 72
Complete the definition:
\int_a^b f(x)dx =
\int_a^b f(x)dx =\lim_{n\to\infty}\Sigma_{i=1}^n f(x_i^*)\Deltax
\int_a^b f(x)dx \text{ represents the net area between a function and the x-axis}
Find the exact value of
\int_{-2}^2\sqrt{4-x^2}dx
\int_{-2}^2\sqrt{4-x^2}dx=2pi
int_0^1 x^2(3-10x^3)^4 dx
Rewrite the integral using u-sub!
int_0^1 x^2(3-10x^3)^4 dx = -1/30 int _3^-7 u^4 du
int (x^-1+e^x+k)dx
int (x^-1+e^x+k)dx = ln|x|+e^x+kx+C
Evaluate
\int_0^pi (2\sin(x)-cos(x))dx
\int_0^pi (2\sin(x)-cos(x))dx=4
Explain the meaning of
Σ_{i=1}^10 (2i+1)
Σ_{i=1}^10 (2i+1) Means "sum the first 10 terms of 2i+1 starting at i=1 . That is,
Σ_{i=1}^10 (2i+1)= (2(1)+1)+(2(2)+1)+...+(2(10)+1)
Julie is a sky diver who executes their jumps from an altitude of 12,500 ft. After they step off the aircraft, they immediately starts falling. Given that the acceleration due to gravity is -32 ft/s2, provide a function s(t) that models Julie's position at time t .
s(t)=-16t^2+12500
int 3/{5y+4}dy
Evaluate the integral using u-sub!
int 3/{5y+4}dy = 3/5 ln|5y+4| +C
\int (sin(x)+cos(x)+sec^2(x)+csc(x)cot(x))dx
\int (sin(x)+cos(x)+sec^2(x)+csc(x)cot(x))dx
=-cos(x)+sin(x)+tan(x)-csc(x)+C
Evaluate
d/dx \int_3^x e^{-t^2}dt
d/dx \int_3^x e^{-t^2}dt=e^{-x^2}
True or False?
∫ {4cosx}/sin^2x dx=-4csc(x) + C
Why/why not?
True
∫ {4cosx}/sin^2x dx=-4csc(x) + C
because
d/dx (-4csc(x)+C)=-4csc(x)cot(x)
=-4(1/sin(x))(cos(x)/sin(x))
=-4cos(x)/sin^2(x)
Kathy can skate at a velocity of 10+cos(pi t) feet per second. What is Kathy's total displacement in the first 5 seconds of skating?
Kathy's displacement is
int_0^5 (10+cos(pi t))dt= 50 feet
int 1/sqrt {1-4x^2}dx
Evaluate using u-sub!
int 1/sqrt {1-4x^2}dx = 1/2 arcsin(2x)+C
\int (3/{1+x^2}+1/sqrt{1-x^2})dx
\int (3/{1+x^2}+1/sqrt{1-x^2})dx= 3arctan(x)+arcsin(x)+C
Given
A(x)= \int_{3x^2}^a cos^2(t)dt,
find A'(x) .
A'(x)=d/dx \int_{3x^2}^a cos^2(t)dt=-cos^2(3x^2)(6x)