Check the solution to "Evaluate the definite integral. ∫−443x4dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Frank Guyton

Answered question

2021-12-31

Evaluate the definite integral.

\int_{- 4}^{4} 3^{\frac{x}{4}} d x

Answer & Explanation

John Koga

Beginner2022-01-01Added 33 answers

The given integral is

\int_{- 4}^{4} 3^{\frac{x}{4}} d x

Substituting

\frac{x}{4}

as t.
Therefore

\frac{d x}{4} = d t

Evaluating the integral:

\int_{- 4}^{4} 3^{\frac{x}{4}} d x = 4 \int_{- 1}^{1} 3^{t} d t

= 4 {[\frac{3^{t}}{\ln 3}]}_{- 1}^{1}

= 4 [\frac{3}{\ln 3} - \frac{3^{- 1}}{\ln 3}]

= \frac{4}{\ln 3} (3 - \frac{1}{3})

= \frac{32}{3 \ln 3}

kaluitagf

Beginner2022-01-02Added 38 answers

\int_{- 4}^{4} 3^{\frac{x}{4}} d x

\int 3^{\frac{x}{4}} d x

\int 3^{t} \cdot 4 d t

4 \cdot \int 3^{t} d t

4 \cdot \frac{3^{t}}{\ln (3)}

4 \cdot \frac{3^{\frac{x}{4}}}{\ln (3)}

Calculate

\frac{4 \cdot 3^{\frac{x}{4}}}{\ln (3)}

Return the limits

\frac{4 \cdot 3^{\frac{x}{4}}}{\ln (3)} ∣_{- 4}^{4}

Calculate the expression

\frac{4 \cdot 3^{\frac{4}{4}}}{\ln (3)} - \frac{4 \cdot 3^{\frac{- 4}{4}}}{\ln (3)}

Simplify
Answer:

\frac{32}{3 \ln (3)}

Vasquez

Expert2022-01-07Added 669 answers

$\int_{- 4}^{4} 3^{x / 4} d x$
This is a tabular integral:
$\int 3^{x / 4} d x = \frac{4 * 3^{x / 4}}{\ln (3)} + C$
Let's calculate a definite integral:
$\begin{matrix} G i v e n : \\ \int_{- 4}^{4} 3^{x / 4} d x = (\frac{4 * 3^{x / 4}}{\ln (3)}) |_{- 4}^{4} \\ F (4) = \frac{12}{\ln (3)} \\ F (- 4) = \frac{4}{3 * \ln (3)} \\ I = \frac{12}{\ln (3)} 0 (\frac{4}{3 * \ln (3)}) = \frac{32}{3 * \ln (3)} \end{matrix}$

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