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

Evaluate the integrals $$\displaystyle\int{e}^{{{t}}}{\cos{{\left({3}{e}^{{{t}}}-{2}\right)}}}{\left.{d}{t}\right.}$$

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Nicole Keller
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
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.
$$\displaystyle{\frac{{{\sin{{\left({3}{e}^{{{t}}}-{2}\right)}}}}}{{{3}}}}$$
$$\displaystyle{\frac{{{\sin{{\left({3}{e}^{{{t}}}-{2}\right)}}}}}{{{3}}}}+{C}$$