Calculus/Integration techniques/Recognizing Derivatives and the Substitution Rule/Solutions

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1.

Let

Then

2.

Let

Then

3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): \int {\frac {17\sin(x)}{\cos(x)}}dx

Let

Then

4.

Let

Then

5.

Let

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \begin{align}\int_{0}^{1}-\frac{10}{(-5x-32)^{4}}dx&=-10\int_{u(0)}^{u(1)}\frac{-du}{5u^{4}}\\ &=-2\frac{1}{3u^{3}}\Biggr|_{u(0)}^{u(1)}\\ &=-2\frac{1}{3(-5x-32)^{3}}\Biggr|_{0}^{1}\\ &=-\frac{2}{3}(\frac{1}{(-5-32)^{3}}-\frac{1}{(-32)^{3}})\\ &=-\frac{2}{3}(\frac{1}{(-37)^{3}}+\frac{1}{(32)^{3}})\\ &=\frac{2}{3}(\frac{1}{37^{3}}-\frac{1}{32^{3}})\\ &=\frac{2}{3}\cdot\frac{32^{3}-37^{3}}{32^{3}\cdot37^{3}}\\ &=\frac{2}{3}\cdot\frac{32^{3}-37^{3}}{2^{15}\cdot37^{3}}\\ &=\frac{32^{3}-37^{3}}{2^{14}\cdot3\cdot37^{3}}\\ &=\mathbf{-\frac{17885}{2489696256}}\end{align}}
6.

Let

Then