Question: "Evaluate the following integrals. ∫e3−4xdx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Ashley Bell

Answered question

2021-12-31

Evaluate the following integrals.

\int e^{3 - 4 x} d x

Answer & Explanation

Bubich13

Beginner2022-01-01Added 36 answers

Step 1
Given integral:

\int e^{3 - 4 x} d x

Substitute t=3-4x and differentiate it w.r.t "x"

\frac{d t}{d x} = \frac{d}{d x} (3 - 4 x)

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

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

Therefore, the given integral becomes,

\frac{- 1}{4} \int e^{t} d t

Step 2
We know that

\int e^{x} d x = e^{x} + c

Where, "c" is integration constant.

\Rightarrow \frac{- 1}{4} e^{t} + c

Substitute t=3-4x in above equation, we get

\Rightarrow \frac{- 1}{4} e^{(3 - 4 x)} + c

\Rightarrow \frac{- e^{(3 - 4 x)}}{4} + c

Bubich13

Beginner2022-01-02Added 36 answers

Given

\int e^{3 - 4 x} d x

\int - \frac{e^{t}}{4} d t

- \frac{1}{4} \cdot \int e^{t} d t

Solve

- \frac{1}{4} e^{t}

- \frac{1}{4} e^{3 - 4 x}

- \frac{e^{3 - 4 x}}{4}

Answer:

- \frac{e^{3 - 4 x}}{4} + C

Vasquez

Expert2022-01-07Added 669 answers

$\begin{matrix} \int e^{3 - 4 x} d x \\ = - \frac{1}{4} \int e^{u} d u \\ \int e^{u} d u \\ = e^{u} \\ - \frac{1}{4} \int e^{u} d u \\ = - \frac{e^{u}}{4} \\ u = 3 - 4 x : \\ = - \frac{e^{3 - 4 x}}{4} \\ S o l u t i o n : \\ = - \frac{e^{3 - 4 x}}{4} + C \end{matrix}$

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