Fundamental Theorem of Calculus Practice
MathSci Problems Fundamental Theorem of Calculus Practice Questions ddx ∫0x2cos(t) dt= ?frac{d}{dx},int_{0}^{x^{2}} cos(t),dt = ? ddx ∫g(x)f(x)h(t) dt= ?frac{d}{dx},int_{g(x)}^{f(x)} h(t),dt = ? ∂∂y ∫x2y ysinxarctan(t) dt= ?frac{partial}{partial y},int_{x^{2}y}^{,ysin x}arctan(t),dt = ? ∂∂x ∫ f(x,y) g(x,y)h(t) dt= ?frac{partial}{partial x},int_{,f(x,y)}^{,g(x,y)} h(t),dt = ? ∂∂xn ∫ g(x1,…,xn) f(x1,…,xn)h(t) dt= Solutions 1)Here is how I like to think of the fundamental theorem of calculus: ∫0g(x)f(t) dt=F(g(x))−F(0)int_0^{g(x)} f(t),dt=F(g(x))-F(0) Notice that F(g(x))F(g(x)) is a function, […]
Integrals Conceptual Practice
MathSci Problems Integrals Conceptual Practice Practice Suppose ∫abf(x) dx=1.Find ∫ba2f(x) dx.text{Suppose } int_{a}^{b} f(x),dx = 1. quad text{Find } int_{b}^{a} 2f(x),dx. ∫0 dx = ?int 0,dx ;=; ? ∫ab0 dx = ?int_{a}^{b} 0,dx ;=; ? Suppose ∫02f(x) dx=5.Find ∫20(3f(x)+4) dx.text{Suppose } int_{0}^{2} f(x),dx = 5. quad text{Find } int_{2}^{0} big(3f(x)+4big),dx. Solutions 1)Use linearity and reversing limits: ∫abkf(x) dx=k∫abf(x) dx,∫abf(x) dx=−∫baf(x) dx.int_{a}^{b} kf(x),dx = kint_{a}^{b} f(x),dx, qquad int_{a}^{b} f(x),dx […]
U-Substitution Practice Problems
MathSci Problems U-Substitution Practice Problems Questions ∫cosx esinx dxint cos x, e^{sin x},dx ∫2ax+bax2+bx+c dx(a≠0)int frac{2ax+b}{ax^2+bx+c},dx quad(aneq 0) ∫cosx (1+sin2x) dxint cos x,(1+sin^2 x),dx ∫cosx cos5(sinx) dxint cos x;cos^{5}(sin x),dx ∫dxxlnx(x>0, x≠1)int frac{dx}{xln x} qquad(x>0, xneq 1) ∫x1+x2 dxint frac{x}{1+x^2},dx ∫x 1+x2 dxint frac{x}{sqrt{,1+x^2,}},dx ∫ex1+ex dxint frac{e^{x}}{1+e^{x}},dx ∫cosx a+sinx dx(a∈R)int frac{cos x}{,a+sin x,},dx quad(ainmathbb{R}) ∫kx+m(kx+m)2+c dx(k≠0)int frac{kx+m}{(kx+m)^2+c},dx quad(kneq 0) […]