5.3: The Fundamental Theorem of Calculus
- Page ID
- 144294
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Use the first part of the fundamental theorem of calculus to find \(\displaystyle \frac{d}{dx} \int_1^x e^{-t^2}\ dt\).
- Use the first part of the fundamental theorem of calculus to find \(\displaystyle \frac{d}{dx} \int_3^x \sqrt{9 - t^2}\ dt\).
- Use the first part of the fundamental theorem of calculus to find \(\displaystyle \frac{d}{dx} \int_{-5}^x \dfrac{\cos s}{1 + s^2}\ ds\).
- Use the first part of the fundamental theorem of calculus to find \(\displaystyle \frac{d}{dx} \int_0^{2x} \tan(t^2)\ dt\).
- Use the first part of the fundamental theorem of calculus to find \(\displaystyle \frac{d}{dx} \int_0^{\sin x} \frac{1}{\sqrt{1 - y^2}}\ dy\).
- Use the first part of the fundamental theorem of calculus to find \(\displaystyle \frac{d}{dx} \int_x^3 \ln(t)\ dt\).
- Use the first part of the fundamental theorem of calculus to find \(\displaystyle \frac{d}{dx} \int_{1 - x}^{x^2} (2t^2 - 3t)\ dt\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{-1}^2 3x^2\ dx\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{-2}^3 (x^2 + 3x - 5)\ dx\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{-2}^3 (t+2)(t-3)\ dt\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{1}^4 \left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)\ dx\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{1}^2 \dfrac{2z^4 + z^3 - 3z}{z^3}\ dz\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{0}^{2\pi} \cos \theta\ d\theta\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{0}^{\pi/2} \sin \theta\ d\theta\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{0}^{\pi/4} \sec^2 \theta\ d\theta\).
- Use the second part of the fundamental theorem of calculus to calculate \(\displaystyle \int_{\pi/6}^{\pi/4} \sec \theta \tan \theta\ d\theta\).