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Use the integral test to determine whether the series {eq}displaystyle sum_{n=1}^{infty}n^5e^{-n^6} {/eq} is convergent or divergent. Evaluate the integral, {eq}displaystyle.

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5e^2x – e^2x {/eq}+ 1) dx from limit 0 to log 2. and the integration by parts formula {eq}int f(x)g(x)dx=f(x)int g(x)dx-int left ( f'(x)int g(x)dx right )dx {/eq}.

In particular, if we have the nonhomogeneous d.e. {eq}f(D)y = R(x) {/eq} the auxiliary equation and solutions for the homogeneous analogue is given by {eq}f(m) = 0 {/eq}, {eq}m_1,m_2,ldots,m_n, {/eq}.

Substitution method is the convenient method in the section of integral calculus. Using this method. int 10xe^{x^2 + 1} dx = int 5e^{t} dt\ = 5 e^{t} \ = 5 e^{x^2 + 1} {/eq}.

The linearity of Integration is similar to the Sum rule of integration with an extended version. Let {eq}displaystyle{ f }{/eq} and {eq}displaystyle{ g }{/eq} be two differentiable functions and.

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The limit of a vector function is calculated componentwise. So given a vector function {eq}vec r (t) = left langle x(t), ; y(t), ; z(t) right rangle {/eq} and a parameter {eq}t_0 {/eq} inside.

begin{align} int frac{5e^3 sqrt{5}}{sqrt{y}} dy &= 5e^3 sqrt{5}int int y^{-frac{1}{2}}dy\ &= 5e^3 sqrt{5}frac{1}{frac{1}{2}} y^{frac{1}{2}} +C\ &= 10e^3 sqrt{5y} +C end{align}.

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y”+ay’+y = f(x)\ y”+ay’+by\= a(y_{h. we will we the given differential equation equal to the first term and use substitution. {eq}y”+y’=5e^x \ y = a_{0} e^x hspace {1cm} y’ = a_{0}e^x.

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