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Even / Odd Functions
2021-07-31
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The graph of an even function is symmetric about the y-axis. Therefore if we have plotted the graph of for , w […]

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Integration by Parts
2021-07-28
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Integration by parts is a formula for transforming the integral of the multiplication of two functions. It may […]

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Substitution Rule / Rational Function of e^{x}
2021-07-27
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Assume that is a rational function of X. Then, in the case of , the substitution will produce an integral of a […]

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Logarithmic Functions
2021-07-26
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Let a>0 and . Then, the inverse function of the exponential function is defined as the logarithmic function […]

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Angeles between Vectors
2021-07-26
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If and are nonzero vectors in either or , there is a relationship between their inner product and the angle be […]

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Proof | Property of Inner Product (3)
2021-07-24
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Proof \begin{eqnarray*}\mathbf{a}\cdot (\mathbf{b}+\mathbf{c})&=&[a_{1} \ a_{2}\ \cdots\ a_{n}]\begin{ […]

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Proof | Property of Inner Products (1)
2021-07-24
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Proof \begin{eqnarray*}&&\mathbf{a}\cdot \mathbf{a}=a_{1}a_{1}+a_{2}a_{2}+ \cdots +a_{n}a_{n}\\&&a […]

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Substitution Rule / Irrational Functions
2021-07-24
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The integrals of irrational functions are rarely represented by well-known functions. We can calculate only if […]

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Proof | Property of Inverse Matrix (4)
2021-07-20
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Proof By the property of the determinant, we have $$\mathrm{det }(AB)=\mathrm{det }A\ \mathrm{det }B$$ Then we […]

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Proof | Property of Inverse Matrix (3)
2021-07-20
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Proof We use the property of the transposed matrix : $$(AB)^{t}=B^{t}A^{t}$$ Then we have \begin{eqnarray*}A^{ […]

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