Worksheet

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\lhead{\textbf{MATH 152 --- Calculus II}}
\chead{\textbf{Worksheet 7: Techniques of Integration}}
\rhead{\textbf{Spring 2026}}
\lfoot{Instructor: Prof.\ M.\ Reyes}
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\begin{center}
  {\Large\bfseries Worksheet 7: Techniques of Integration}\\[6pt]
  \begin{tabularx}{0.8\textwidth}{@{}X X@{}}
    \textbf{Name:} \hrulefill & \textbf{Section:} \hrulefill \\[6pt]
    \textbf{Date:} \hrulefill & \textbf{Score:} \quad\underline{\hspace{1cm}} / 50 \\
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{\small\textit{Show all work for full credit. Simplify your answers where possible. You may use a calculator for arithmetic only.}}

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\textbf{\large Part A: Integration by Substitution} \hfill (15 points)

\problem Evaluate the following integrals using $u$-substitution. \textit{(5 pts each)}

\begin{enumerate}[label=(\alph*),leftmargin=*]
  \item $\displaystyle\int x^2 \sqrt{x^3 + 1}\, dx$

  \answerspace{4cm}

  \item $\displaystyle\int \frac{\cos(\ln x)}{x}\, dx$

  \answerspace{4cm}

  \item $\displaystyle\int_0^{\pi/4} \tan x \sec^2 x\, dx$

  \answerspace{4cm}
\end{enumerate}

\textbf{\large Part B: Integration by Parts} \hfill (15 points)

\problem Evaluate the following integrals using integration by parts. \textit{(5 pts each)}

\begin{enumerate}[label=(\alph*),leftmargin=*]
  \item $\displaystyle\int x e^{3x}\, dx$

  \answerspace{4cm}

  \item $\displaystyle\int x^2 \sin x\, dx$

  \answerspace{5cm}

  \item $\displaystyle\int \ln(x^2 + 1)\, dx$ \quad \textit{(Hint: let $u = \ln(x^2 + 1)$ and $dv = dx$)}

  \answerspace{5cm}
\end{enumerate}

\textbf{\large Part C: Trigonometric Integrals} \hfill (10 points)

\problem Evaluate the following integrals. \textit{(5 pts each)}

\begin{enumerate}[label=(\alph*),leftmargin=*]
  \item $\displaystyle\int \sin^3 x \cos^2 x\, dx$

  \answerspace{4.5cm}

  \item $\displaystyle\int \sec^4 x\, dx$

  \answerspace{4.5cm}
\end{enumerate}

\textbf{\large Part D: Partial Fractions} \hfill (10 points)

\problem Evaluate using partial fraction decomposition. \textit{(5 pts each)}

\begin{enumerate}[label=(\alph*),leftmargin=*]
  \item $\displaystyle\int \frac{3x + 5}{(x+1)(x+3)}\, dx$

  \answerspace{5cm}

  \item $\displaystyle\int \frac{2x^2 + x + 4}{x^3 + 4x}\, dx$ \quad \textit{(Hint: Factor the denominator first.)}

  \answerspace{5cm}
\end{enumerate}

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  \textit{--- End of Worksheet ---}
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