Evaluate the integrals \int e^{t}\cos (3e^{t}-2)dt

sputavanomr 2021-11-23 Answered
Evaluate the integrals \(\displaystyle\int{e}^{{{t}}}{\cos{{\left({3}{e}^{{{t}}}-{2}\right)}}}{\left.{d}{t}\right.}\)

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Expert Answer

Nicole Keller
Answered 2021-11-24 Author has 46 answers
Step 1
We have to evaluate the integral:
\(\displaystyle\int{e}^{{{t}}}{\cos{{\left({3}{e}^{{{t}}}-{2}\right)}}}{\left.{d}{t}\right.}\)
In this case we should use substitution method since derivatives of one function is in the integral.
So let \(\displaystyle{x}={3}{e}^{{{t}}}-{2}\)
differentiating both sides with respect to 't', we get
\(\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={\frac{{{d}{\left({3}{e}^{{{t}}}-{2}\right)}}}{{{\left.{d}{t}\right.}}}}\)
\(\displaystyle={3}{\frac{{{d}{e}^{{{t}}}}}{{{\left.{d}{t}\right.}}}}-{\frac{{{d}{2}}}{{{\left.{d}{t}\right.}}}}\)
\(\displaystyle={3}{e}^{{{t}}}-{0}\)
\(\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={3}{e}^{{{t}}}\)
\(\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{3}}}}={e}^{{{t}}}{\left.{d}{t}\right.}\)
Step 2
Substituting above values, we get
\(\displaystyle\int{e}^{{{t}}}{\cos{{\left({3}{e}^{{{t}}}-{2}\right)}}}{\left.{d}{t}\right.}=\int{\cos{{\left({3}{e}^{{{t}}}-{2}\right)}}}{e}^{{{t}}}{\left.{d}{t}\right.}\)
\(\displaystyle={\frac{{{1}}}{{{3}}}}\int{\cos{{x}}}{\left.{d}{x}\right.}\)
\(\displaystyle={\frac{{{1}}}{{{3}}}}{\left({\sin{{x}}}\right)}+{C}\)
Where, C is an arbitrary constant.
Hence, value of given integral is \(\displaystyle{\frac{{{1}}}{{{3}}}}{\sin{{x}}}+{C}\).
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Prioned
Answered 2021-11-25 Author has 532 answers
Step 1: Use Integration by Substitution.
Let \(\displaystyle{u}={3}{e}^{{{t}}}-{2},{d}{u}={3}{e}^{{{t}}}{\left.{d}{t}\right.},\ {t}{h}{e}{n}\ {e}^{{{t}}}{\left.{d}{t}\right.}={\frac{{{1}}}{{{3}}}}{d}{u}\)
Step 2: Using u and du above, rewrite \(\displaystyle\int{e}^{{{t}}}{\cos{{\left({3}{e}^{{{t}}}-{2}\right)}}}{\left.{d}{t}\right.}\).
\(\displaystyle\int{\frac{{{\cos{{u}}}}}{{{3}}}}{d}{u}\)
Step 3: Use Constant Factor Rule: \(\displaystyle\int{c}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle{\frac{{{1}}}{{{3}}}}\int{\cos{{u}}}{d}{u}\)
Step 4: Use Trigonometric Integration: the integral of \(\displaystyle{\cos{{u}}}\ {i}{s}\ {\sin{{u}}}\).
\(\displaystyle{\frac{{{\sin{{u}}}}}{{{3}}}}\)
Step 5: Substitute \(\displaystyle{u}={3}{e}^{{{t}}}-{2}\) back into the original integral.
\(\displaystyle{\frac{{{\sin{{\left({3}{e}^{{{t}}}-{2}\right)}}}}}{{{3}}}}\)
Step 6: Add constant.
\(\displaystyle{\frac{{{\sin{{\left({3}{e}^{{{t}}}-{2}\right)}}}}}{{{3}}}}+{C}\)
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