# Given \int (x^{5}-3x^{4}+6x^{3}+3)dx, evaluate the indefinite integral.

Given $$\displaystyle\int{\left({x}^{{{5}}}-{3}{x}^{{{4}}}+{6}{x}^{{{3}}}+{3}\right)}{\left.{d}{x}\right.}$$, evaluate the indefinite integral.

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Beverly Smith
Step 1
The given integral is $$\displaystyle\int{\left({x}^{{{5}}}-{3}{x}^{{{4}}}+{6}{x}^{{{3}}}+{3}\right)}{\left.{d}{x}\right.}$$.
obtain the integral value as follows.
$$\displaystyle\int{\left({x}^{{{5}}}-{3}{x}^{{{4}}}+{6}{x}^{{{3}}}+{3}\right)}{\left.{d}{x}\right.}=\int{x}^{{{5}}}{\left.{d}{x}\right.}-\int{3}{x}^{{{4}}}{\left.{d}{x}\right.}+\int{6}{x}^{{{3}}}{\left.{d}{x}\right.}+\int{3}{\left.{d}{x}\right.}$$
$$\displaystyle={\frac{{{x}^{{{6}}}}}{{{6}}}}-{3}{\left({\frac{{{x}^{{{5}}}}}{{{5}}}}\right)}+{6}{\left({\frac{{{x}^{{{4}}}}}{{{4}}}}\right)}+{3}{x}+{C}$$
$$\displaystyle={\frac{{{x}^{{{6}}}}}{{{6}}}}-{\frac{{{3}{x}^{{{5}}}}}{{{5}}}}+{\frac{{{3}{x}^{{{4}}}}}{{{2}}}}+{3}{x}+{C}$$
Thus, the value of the integral is $$\displaystyle{\frac{{{x}^{{{6}}}}}{{{6}}}}-{\frac{{{3}{x}^{{{5}}}}}{{{5}}}}+{\frac{{{3}{x}^{{{4}}}}}{{{2}}}}+{3}{x}$$
Step 2
The value of the given integral is $$\displaystyle{\left(\frac{{x}^{{{6}}}}{{6}}\right)}-{3}{\left(\frac{{x}^{{{5}}}}{{5}}\right)}+{3}{\left(\frac{{x}^{{{4}}}}{{2}}\right)}+{3}{x}$$.
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ambarakaq8
Given integral.
$$\displaystyle\int{x}^{{{5}}}-{3}{x}^{{{4}}}+{6}{x}^{{{3}}}+{3}{\left.{d}{x}\right.}$$
$$\displaystyle\int{x}^{{{5}}}{\left.{d}{x}\right.}-\int{3}{x}^{{{4}}}{\left.{d}{x}\right.}+\int{6}{x}^{{{3}}}{\left.{d}{x}\right.}+\int{3}{\left.{d}{x}\right.}$$
Evaluate
$$\displaystyle{\frac{{{x}^{{{6}}}}}{{{6}}}}-{\frac{{{3}{x}^{{{5}}}}}{{{5}}}}+{\frac{{{3}{x}^{{{4}}}}}{{{2}}}}+{3}{x}$$
$$\displaystyle{\frac{{{x}^{{{6}}}}}{{{6}}}}-{\frac{{{3}{x}^{{{5}}}}}{{{5}}}}+{\frac{{{3}{x}^{{{4}}}}}{{{2}}}}+{3}{x}+{C}$$
karton

$$I=\int (x^{5}-3x^{4}+6x^{3}+3)dx \\\text{we know that} \\\int x^{n}dx=\frac{x^{n+1}}{n+1} \\I=\frac{x^{5+1}}{5+1}-\frac{3x^{4+1}}{4+1}+\frac{6x^{3+1}}{3+1}+3x \\I=\frac{x^{6}}{6}-\frac{3x^{5}}{5}+\frac{6x^{4}}{4}+3x \\I=\frac{x^{6}}{6}-\frac{3}{5}x^{5}+\frac{3}{2}x^{4}+3x$$