Question 231712
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ F(x)\ =\ \int\ x^4\ +\ 3x^2\,dx]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \int\ f(x)\ +\ g(x)\,dx\ =\ \int\ f(x)\,dx\ +\ \int\ g(x)\,dx]


so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ F(x)\ =\ \int\ x^4\,dx\ +\ \int\ 3x^2\,dx]


Power Rule:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \int\ x^n\,dx\ =\ \frac{x^{\small{n+1}}\LARGE}{n+1}\ +\ C]


Also:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \int\ af(x)\,dx\ =\ a\int\ f(x)\,dx]


so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ F(x)\ =\ \frac{x^5}{5}\ +\ C_1\ +\ 3\frac{x^3}{3}\ +\ C_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ F(x)\ =\ \frac{x^5}{5}\ +\ x^3\ +\ C]


Check:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ F'(x)\ =\ f(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 5\cdot\frac{x^4}{5}\ + 3x^2\ +\ 0\ =\ x^4\ +\ 3x^2]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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