Evaluate the indefinite integral. (Use C for the constant of

tapetivk 2021-11-22 Answered
Evaluate the indefinite integral. (Use C for the constant of integration.)
\(\displaystyle\int{e}^{{{\cos{{\left({9}{t}\right)}}}}}{\sin{{\left({9}{t}\right)}}}{\left.{d}{t}\right.}\)

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Expert Answer

Howell
Answered 2021-11-23 Author has 8515 answers
Step 1
Given integral:
\(\displaystyle\int{e}^{{{\cos{{\left({9}{t}\right)}}}}}{\sin{{\left({9}{t}\right)}}}{\left.{d}{t}\right.}\)
Step 2
Now,
\(\displaystyle\int{e}^{{{\cos{{\left({9}{t}\right)}}}}}{\sin{{\left({9}{t}\right)}}}{\left.{d}{t}\right.}\)
Sustitute:
\(\displaystyle{\cos{{\left({9}{t}\right)}}}={u}\)
Differentiate both sides:
\(\displaystyle\Rightarrow-{\sin{{\left({9}{t}\right)}}}\times{9}{\left.{d}{t}\right.}={d}{u}\ \ \ {\left[\because{\frac{{{d}{\left({\cos{{\left({x}\right)}}}\right)}}}{{{\left.{d}{x}\right.}}}}=-{\sin{{\left({x}\right)}}}\right]}\)
\(\displaystyle\Rightarrow{\sin{{\left({9}{t}\right)}}}{\left.{d}{t}\right.}=-{\frac{{{d}{u}}}{{{9}}}}\)
Now the integral becomes:
\(\displaystyle\int-{\frac{{{1}}}{{{9}}}}{e}^{{{u}}}{d}{u}=-{\frac{{{1}}}{{{9}}}}\int{e}^{{{u}}}{d}{u}\ \ \ {\left[\because\int{e}^{{{x}}}{\left.{d}{x}\right.}={e}^{{{x}}}+{c}\right]}\)
\(\displaystyle=-{\frac{{{1}}}{{{9}}}}{e}^{{{u}}}+{C}\)
Substitute back the value of u:
\(\displaystyle=-{\frac{{{1}}}{{{9}}}}{e}^{{{\cos{{\left({9}{t}\right)}}}}}+{C}\)
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Roger Noah
Answered 2021-11-24 Author has 7902 answers
Step 1: Use Integration by Substitution.
Let \(\displaystyle{u}={\cos{{9}}}{t},{d}{u}=-{9}{\sin{{9}}}{t}{\left.{d}{t}\right.},\ {t}{h}{e}{n}\ {\sin{{9}}}{t}{\left.{d}{t}\right.}=-{\frac{{{1}}}{{{9}}}}{d}{u}\)
Step 2: Using u and du above, rewrite \(\displaystyle\int{e}^{{{\cos{{9}}}{t}}}{\sin{{9}}}{t}{\left.{d}{t}\right.}\).
\(\displaystyle\int-{\frac{{{e}^{{{u}}}}}{{{9}}}}{d}{u}\)
Step 3: Use Constant Factor Rule: \(\displaystyle\int{c}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle-{\frac{{{1}}}{{{9}}}}\int{e}^{{{u}}}{d}{u}\)
Step 4: The integral of \(\displaystyle{e}^{{{x}}}\ {i}{s}\ {e}^{{{x}}}\).
\(\displaystyle-{\frac{{{e}^{{{u}}}}}{{{9}}}}\)
Step 5: Substitute \(\displaystyle{u}={\cos{{9}}}{t}\) back into the original integral.
\(\displaystyle-{\frac{{{e}^{{{\cos{{9}}}{t}}}}}{{{9}}}}\)
Step 6: Add constant.
\(\displaystyle-{\frac{{{e}^{{{\cos{{9}}}{t}}}}}{{{9}}}}+{C}\)
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