1. Find each of the following indefinite integrals.
(a) \( \displaystyle \int (3x^3 - 4\sqrt{x} + 3) \, dx \)
Solution
\(
\begin{aligned}
\int (3x^3 - 4\sqrt{x} + 3) \, dx &= \int \left(3x^3 - 4x^{\frac{1}{2}} + 3\right) \, dx \\
&= \frac{3x^4}{4} - \frac{4x^{\frac{3}{2}}}{\frac{3}{2}} + 3x + C \\
&= \frac{3}{4}x^4 - \frac{8}{3}x^{\frac{3}{2}} + 3x + C
\end{aligned}
\)
(b) \( \displaystyle \int \left(6x^2 - \frac{4}{x^2}\right) \, dx \)
Solution
\(
\begin{aligned}
\int \left(6x^2 - \frac{4}{x^2}\right) \, dx &= \int \left(6x^2 - 4x^{-2}\right) \, dx \\
&= \frac{6x^3}{3} - \frac{4x^{-1}}{-1} + C \\
&= 2x^3 + \frac{4}{x} + C
\end{aligned}
\)
(c) \( \displaystyle \int \left(5 - \frac{1}{\sqrt{x}} + \frac{1}{x^3}\right) \, dx \)
Solution
\(
\begin{aligned}
\int \left(5 - \frac{1}{\sqrt{x}} + \frac{1}{x^3}\right) \, dx &= \int \left(5 - x^{-\frac{1}{2}} + x^{-3}\right) \, dx \\
&= 5x - \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + \frac{x^{-2}}{-2} + C \\
&= 5x - 2\sqrt{x} - \frac{1}{2x^2} + C
\end{aligned}
\)
(d) \( \displaystyle \int \frac{x^4 + 5x}{2x^3} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{x^4 + 5x}{2x^3} \, dx &= \int \left(\frac{x^4}{2x^3} + \frac{5x}{2x^3}\right) \, dx \\
&= \int \left(\frac{1}{2}x + \frac{5}{2}x^{-2}\right) \, dx \\
&= \frac{1}{2}\left(\frac{x^2}{2}\right) + \frac{5}{2}\left(\frac{x^{-1}}{-1}\right) + C \\
&= \frac{1}{4}x^2 - \frac{5}{2x} + C
\end{aligned}
\)
(e) \( \displaystyle \int \frac{3x}{2\sqrt[5]{x^2}} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{3x}{2\sqrt[5]{x^2}} \, dx &= \int \frac{3x}{2x^{\frac{2}{5}}} \, dx \\
&= \int \frac{3}{2} x^{1 - \frac{2}{5}} \, dx \\
&= \int \frac{3}{2} x^{\frac{3}{5}} \, dx \\
&= \frac{3}{2} \left( \frac{x^{\frac{8}{5}}}{\frac{8}{5}} \right) + C \\
&= \frac{15}{16} x^{\frac{8}{5}} + C
\end{aligned}
\)
(f) \( \displaystyle \int \frac{(3x - 1)^2}{5x^4} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{(3x - 1)^2}{5x^4} \, dx &= \int \frac{9x^2 - 6x + 1}{5x^4} \, dx \\
&= \int \left( \frac{9}{5}x^{-2} - \frac{6}{5}x^{-3} + \frac{1}{5}x^{-4} \right) \, dx \\
&= \frac{9}{5}\left(\frac{x^{-1}}{-1}\right) - \frac{6}{5}\left(\frac{x^{-2}}{-2}\right) + \frac{1}{5}\left(\frac{x^{-3}}{-3}\right) + C \\
&= -\frac{9}{5x} + \frac{3}{5x^2} - \frac{1}{15x^3} + C
\end{aligned}
\)
(g) \( \displaystyle \int \frac{3x^7 + x^2}{2\sqrt[3]{x}} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{3x^7 + x^2}{2\sqrt[3]{x}} \, dx &= \int \left( \frac{3x^7}{2x^{\frac{1}{3}}} + \frac{x^2}{2x^{\frac{1}{3}}} \right) \, dx \\
&= \int \left( \frac{3}{2} x^{7 - \frac{1}{3}} + \frac{1}{2} x^{2 - \frac{1}{3}} \right) \, dx \\
&= \int \left( \frac{3}{2} x^{\frac{20}{3}} + \frac{1}{2} x^{\frac{5}{3}} \right) \, dx \\
&= \frac{3}{2} \left( \frac{x^{\frac{23}{3}}}{\frac{23}{3}} \right) + \frac{1}{2} \left( \frac{x^{\frac{8}{3}}}{\frac{8}{3}} \right) + C \\
&= \frac{9}{46} x^{\frac{23}{3}} + \frac{3}{16} x^{\frac{8}{3}} + C
\end{aligned}
\)
(h) \( \displaystyle \int (x - 3\sqrt{x})^2 \, dx \)
Solution
\(
\begin{aligned}
\int (x - 3\sqrt{x})^2 \, dx &= \int \left( x^2 - 6x\sqrt{x} + 9x \right) \, dx \\
&= \int \left( x^2 - 6x^{\frac{3}{2}} + 9x \right) \, dx \\
&= \frac{x^3}{3} - 6 \left( \frac{x^{\frac{5}{2}}}{\frac{5}{2}} \right) + \frac{9x^2}{2} + C \\
&= \frac{1}{3}x^3 - \frac{12}{5}x^{\frac{5}{2}} + \frac{9}{2}x^2 + C
\end{aligned}
\)
(i) \( \displaystyle \int (1 + \sqrt[4]{x})(1 - \sqrt[4]{x}) \, dx \)
Solution
\(
\begin{aligned}
\int (1 + \sqrt[4]{x})(1 - \sqrt[4]{x}) \, dx &= \int \left( 1 - (\sqrt[4]{x})^2 \right) \, dx \\
&= \int \left( 1 - x^{\frac{2}{4}} \right) \, dx \\
&= \int \left( 1 - x^{\frac{1}{2}} \right) \, dx \\
&= x - \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C \\
&= x - \frac{2}{3}x^{\frac{3}{2}} + C
\end{aligned}
\)
(j) \( \displaystyle \int \left(\sqrt[3]{x} + \frac{2}{\sqrt[3]{x}}\right)^2 \, dx \)
Solution
\(
\begin{aligned}
\int \left(\sqrt[3]{x} + \frac{2}{\sqrt[3]{x}}\right)^2 \, dx &= \int \left( x^{\frac{1}{3}} + 2x^{-\frac{1}{3}} \right)^2 \, dx \\
&= \int \left( x^{\frac{2}{3}} + 4(x^{\frac{1}{3}})(x^{-\frac{1}{3}}) + 4x^{-\frac{2}{3}} \right) \, dx \\
&= \int \left( x^{\frac{2}{3}} + 4 + 4x^{-\frac{2}{3}} \right) \, dx \\
&= \frac{x^{\frac{5}{3}}}{\frac{5}{3}} + 4x + 4\left(\frac{x^{\frac{1}{3}}}{\frac{1}{3}}\right) + C \\
&= \frac{3}{5}x^{\frac{5}{3}} + 4x + 12x^{\frac{1}{3}} + C
\end{aligned}
\)
(k) \( \displaystyle \int (x + 1)(x + 4) \, dx \)
Solution
\(
\begin{aligned}
\int (x + 1)(x + 4) \, dx &= \int (x^2 + 5x + 4) \, dx \\
&= \frac{1}{3}x^3 + \frac{5}{2}x^2 + 4x + C
\end{aligned}
\)
(l) \( \displaystyle \int (x - 3)^2 \, dx \)
Solution
\(
\begin{aligned}
\int (x - 3)^2 \, dx &= \int (x^2 - 6x + 9) \, dx \\
&= \frac{1}{3}x^3 - 3x^2 + 9x + C
\end{aligned}
\)
(m) \( \displaystyle \int (2\sqrt{x} - 1)^2 \, dx \)
Solution
\(
\begin{aligned}
\int (2\sqrt{x} - 1)^2 \, dx &= \int \left( 4x - 4\sqrt{x} + 1 \right) \, dx \\
&= \int \left( 4x - 4x^{\frac{1}{2}} + 1 \right) \, dx \\
&= \frac{4x^2}{2} - 4\left(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right) + x + C \\
&= 2x^2 - \frac{8}{3}x^{\frac{3}{2}} + x + C
\end{aligned}
\)
(n) \( \displaystyle \int \sqrt[3]{x}(x^2 + 1) \, dx \)
Solution
\(
\begin{aligned}
\int \sqrt[3]{x}(x^2 + 1) \, dx &= \int x^{\frac{1}{3}}(x^2 + 1) \, dx \\
&= \int \left( x^{\frac{7}{3}} + x^{\frac{1}{3}} \right) \, dx \\
&= \frac{x^{\frac{10}{3}}}{\frac{10}{3}} + \frac{x^{\frac{4}{3}}}{\frac{4}{3}} + C \\
&= \frac{3}{10}x^{\frac{10}{3}} + \frac{3}{4}x^{\frac{4}{3}} + C
\end{aligned}
\)
(o) \( \displaystyle \int \frac{x^2 - 1}{2x^2} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{x^2 - 1}{2x^2} \, dx &= \int \left( \frac{1}{2} - \frac{1}{2x^2} \right) \, dx \\
&= \int \left( \frac{1}{2} - \frac{1}{2}x^{-2} \right) \, dx \\
&= \frac{1}{2}x - \frac{1}{2}\left(\frac{x^{-1}}{-1}\right) + C \\
&= \frac{1}{2}x + \frac{1}{2x} + C
\end{aligned}
\)
(p) \( \displaystyle \int \frac{x^3 + 6}{2x^3} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{x^3 + 6}{2x^3} \, dx &= \int \left( \frac{1}{2} + \frac{3}{x^3} \right) \, dx \\
&= \int \left( \frac{1}{2} + 3x^{-3} \right) \, dx \\
&= \frac{1}{2}x + 3\left(\frac{x^{-2}}{-2}\right) + C \\
&= \frac{1}{2}x - \frac{3}{2x^2} + C
\end{aligned}
\)
(q) \( \displaystyle \int \frac{x^2 + 2\sqrt{x}}{3x} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{x^2 + 2\sqrt{x}}{3x} \, dx &= \int \left( \frac{x}{3} + \frac{2x^{\frac{1}{2}}}{3x} \right) \, dx \\
&= \int \left( \frac{1}{3}x + \frac{2}{3}x^{-\frac{1}{2}} \right) \, dx \\
&= \frac{1}{3}\left(\frac{x^2}{2}\right) + \frac{2}{3}\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right) + C \\
&= \frac{1}{6}x^2 + \frac{4}{3}\sqrt{x} + C
\end{aligned}
\)
(r) \( \displaystyle \int \frac{4x^2 - 3\sqrt{x}}{x} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{4x^2 - 3\sqrt{x}}{x} \, dx &= \int \left( 4x - 3x^{-\frac{1}{2}} \right) \, dx \\
&= \frac{4x^2}{2} - 3\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right) + C \\
&= 2x^2 - 6\sqrt{x} + C
\end{aligned}
\)
(s) \( \displaystyle \int \left(2\sqrt{x} - \frac{3}{x^2\sqrt{x}}\right)^2 \, dx \)
Solution
\(
\begin{aligned}
\int \left(2\sqrt{x} - \frac{3}{x^2\sqrt{x}}\right)^2 \, dx &= \int \left( 2x^{\frac{1}{2}} - 3x^{-\frac{5}{2}} \right)^2 \, dx \\
&= \int \left( 4x - 12(x^{\frac{1}{2}})(x^{-\frac{5}{2}}) + 9x^{-5} \right) \, dx \\
&= \int \left( 4x - 12x^{-2} + 9x^{-5} \right) \, dx \\
&= \frac{4x^2}{2} - 12\left(\frac{x^{-1}}{-1}\right) + 9\left(\frac{x^{-4}}{-4}\right) + C \\
&= 2x^2 + \frac{12}{x} - \frac{9}{4x^4} + C
\end{aligned}
\)
(t) \( \displaystyle \int \frac{(x + 3)(x - 1)}{\sqrt{x}} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{(x + 3)(x - 1)}{\sqrt{x}} \, dx &= \int \frac{x^2 + 2x - 3}{x^{\frac{1}{2}}} \, dx \\
&= \int \left( x^{\frac{3}{2}} + 2x^{\frac{1}{2}} - 3x^{-\frac{1}{2}} \right) \, dx \\
&= \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + 2\left(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right) - 3\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right) + C \\
&= \frac{2}{5}x^{\frac{5}{2}} + \frac{4}{3}x^{\frac{3}{2}} - 6\sqrt{x} + C
\end{aligned}
\)
(u) \( \displaystyle \int \frac{x^4 - 10}{x\sqrt{x}} \, dx \)
Solution
\(
\begin{aligned}
\int \frac{x^4 - 10}{x\sqrt{x}} \, dx &= \int \frac{x^4 - 10}{x^{\frac{3}{2}}} \, dx \\
&= \int \left( x^{\frac{5}{2}} - 10x^{-\frac{3}{2}} \right) \, dx \\
&= \frac{x^{\frac{7}{2}}}{\frac{7}{2}} - 10\left(\frac{x^{-\frac{1}{2}}}{-\frac{1}{2}}\right) + C \\
&= \frac{2}{7}x^{\frac{7}{2}} + 20x^{-\frac{1}{2}} + C \\
&= \frac{2}{7}x^{\frac{7}{2}} + \frac{20}{\sqrt{x}} + C
\end{aligned}
\)
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