The following results illustrate the need of integration: 1. Trigonometric identity:cos2(x)=1+cos(2x)2. 2. Move the constant out:∫b⋅f(x)dx=b⋅∫f(x)dx. 3. Common integration:∫cos(u)du=sin(u). 4. The sum rule:∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx...
= 2x3+ C Sum Rule Example: What is∫(cos x + x) dx ? Use the Sum Rule: ∫(cos x + x) dx =∫cos x dx +∫x dx Work out the integral of each (using table above): = sin x + x2/2 + C Difference Rule Example: What is∫(ew− 3) dw ?
The integral rules are used to perform the integral easily. In fact, the integral of a function f(x) is a function F(x) such that d/dx (F(x)) = f(x). For example, d/dx (x2) = 2x and so ∫ 2x dx = x2+ C. i.e., the integration is the reverse process of differentiat...
{eq}\begin{align} \frac{d}{{dx}}\left( {{x^n}} \right) &= n{x^{n - 1}},\\ \int {\sin ax} dx &= - \frac{{\cos ax}}{a} + c,\\ \int {\cos ax} dx &= \frac{{\sin ax}}{a} + c, \end{align} {/eq} where {eq}c {/eq} is a constant ...
Use integration by parts to find the integral of:[Hint: In (7) write lnx as 1lnx and in (9) write arctanxdn as 1arctanax.](1)xe^x(2)xsinx(3)x^2lnx(4)xsin3x(5)xcos2x(6)xsec^2x(7)lnx(8)(lnx)^2(9)arctanxdn ...
In (9) write arctan x as 1arctan x. (1)x^x (2)x sin x (3)x^2ln x (4)x sin 3x (5)x cos 2x (6)x sec^2x (7)ln x (8)(ln x)^2 (9)arctan x 相关知识点: 试题来源: 解析 (1)x^x-^x+c (2)-x cos x +sin x +c (3)13x^3ln x-19x^3+c (4)-13x...
Use integration by parts to find the indefinite integral: \int x^2\sin x dx Evaluate using Integration by Parts. (Use C for the constant of integration.) \int e^{7x} \cos(3x) dx Use integration by parts to evaluate the indefinite integral: integral ...
Portions of this entry contributed by Jon Michael Smith Explore with Wolfram|AlphaMore things to try: 10^39 differential equations sin 2x limit tan(t) as t->pi/2 from the left ReferencesFowler, M. "A New Numerical Method for Simulation." Simulation 6, 90-92, Feb. 1976.Smith, J. M...
x=\infty, \theta=\frac{\pi}{2},dx=\sec^2 \theta d\theta,x+\sqrt{1+x^2}=\sec\theta+\tan\theta x+\sqrt{1+x^2}=\frac{\sin \theta+1}{\cos\theta}, (x+\sqrt{1+x^2})^3=(1+\sin\theta)^3\sec^3\theta \int _0^\frac{\pi}{2} \frac{1}{(1+\sin\theta)^3}...
x2xdx 2x sin xdxx sin x x sin x= dx = sin x x sin x x(sin x + x cos x)dx NO!dx x +x cos x +x 2 cos x6There are some simple integrals where little choice is available: knowing which of a large numbertechniques to use is crucial.Example: ln xdx obviously requiresdxu =...