那么,哪个函数是u(x),哪个函数是v(x)呢? 对比公式,u(x)应该是在微分符号前面的一个函数,而v(x)应该是在微分符号后面的,而xsinx dx这个式子中,有两个函数都在微分符号的前面。 所以,我们要做的第一步,就是把其中的一个函数...
局限性:仅适用于u求导后归零的情况(如多项式),不适用于循环结构(如∫eˣsinx dx)。 依赖经验:需正确选择u和dv,否则可能增加计算复杂度。应用场景多项式与指数/三角函数乘积的积分 如∫x⁴cos(2x)dx、∫x²e⁻ˣdx等。 工程与物理学问题 振动分析、电路微分方程求解...
Integral ∫10(x(1−x))ndndnx(logx⋅log(1−x))dx∫01(x(1−x))ndndnx(logx⋅log(1−x))dx 7 Calculating ∫π/20(xsinx)ndx∫0π/2(xsinx)ndx 0 Prove ∫∞0(∏nk=1sin(t/kx)t/kx)dt=π2∫0∞(∏k=1nsin(t/kx)t/kx)dt=...
{eq}\displaystyle\int x^n\ dx=\dfrac{x^{n+1}}{n+1}+c\\\ \displaystyle\int sinx\ dx=-cosx+c\\\ \displaystyle\int cosx\ dx=sinx+c\\\ \displaystyle\int \dfrac{1}{x}dx=ln\left | x \right |+c\\\ {/eq} Answer and Expl...
\int tan^3(x) sec^6(x) dx Evaluate the integral. 5\int\sqrt{1-sinx} dx ( use C for constant of integration) Evaluate the integral. (Use C for the constant of integration.) \int\frac{3dx}{x(x^{4}+2)} Evaluate the integral. (Use C for the...
Prove that limn→∞∫30(√sinxn+x+1)dxn→∞∫03(sinxn+x+1)dx exists and evaluate it 1 How to construct a counterexample to limn→∞∫10fn(x)dx=∫10f(x)dxlimn→∞∫01fn(x)dx=∫01f(x)dx 19 On the integral ∫π0sin(x+sin(x+sin(x+⋯)))dx∫0...
原公式:(uv)'=u'v+uv'求导公式 :d(uv)/dx=(du/dx)v+u(dv/dx)写成全微分形式就成为:d(uv)=vdu+udv 移项后,成为:udv=d(uv) -vdu两边积分得到:/udv=uv-/vdu在传统的微积分教材里分部积分法通常写成不定积分形式: /v(x)u'(x)dx=v(x)u(x)-/v'(x)u(x)dx 例:/xcosxdx=xsinx-/sinxdx...
4.\int\frac{dx}{sinx(2+cosx)} \begin{align} {\frak{Solve}}:&\int\frac{dx}{sinx(2+cosx)} \\ &=\int \frac{d(cosx)}{(cos^{2}x-1)(2+cosx)} \\ &=\frac{1}{3}\int \frac{1}{cosx+2}d(cosx)-\frac{1}{6}\int \frac{1}{1-cosx}d(cosx)-\frac{1}{2}\int \frac...
sin(x) = 2t/(1 +t2) cos(x) = (1 –t2) / (1 +t2) dx= 2dt/(1 +t2). This means that the trick can convert an integral containing any rational combination of trig functions into one involving only rational functions, which then can (in principle) be integrated in closed form usin...
dx •∫ 10xe3x dx •∫(x2 + 5)e-2xdx 1Math 152, c Benjamin Aurispa•∫x7e-x4dx Integrals of the general form∫xm cosx dx or∫xm sinx dx •∫3xsin 2x dx •∫x5 cos(x2) dx 2Math 152, c Benjamin AurispaIntegrals of the general form∫xm lnx dx •∫lnx dx •∫...