Step by step solution to "Evaluate the following integrals. ∫ex4ex+6dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Teddy Dillard

Answered question

2021-12-11

Evaluate the following integrals.

\int \frac{e^{x}}{4 e^{x} + 6} d x

Answer & Explanation

veiga34

Beginner2021-12-12Added 32 answers

Step 1
To evaluate: $\int \frac{e^{x}}{4 e^{x} + 6} d x$
Solution:
Let substitute $t = 4 e^{x} + 6$
Differentiating both sides,
$\frac{d t}{d x} = \frac{d}{d x} (4 e^{x} + 6)$
$\frac{d t}{d x} = 4 e^{x} + 0$
$\frac{d t}{d x} = 4 e^{x}$
$\frac{d t}{4 e^{x}} = d x$
Step 2
Substituting the values in given integral,
$\int \frac{e^{x}}{4 e^{x} + 6} d x = \int \frac{e^{x}}{t} \cdot \frac{d t}{4 e^{x}}$
$= \frac{1}{4} \int \frac{1}{t} d t$
$= \frac{1}{4} \ln | t | + C$
Put back $t = 4 e^{x} + 6$
$\int \frac{e^{x}}{4 e^{x} + 6} d x = \frac{1}{4} \ln | 4 e^{x} + 6 | + C$
Hence, required answer is $\frac{1}{4} \ln | 4 e^{x} + 6 | + C$ .

Mary Herrera

Beginner2021-12-13Added 37 answers

$\int \frac{e^{x}}{4 e^{x} + 6} d x$
We put the expression $\exp (x)$ under the differential sign, i.e.:
$e^{x} d x = d (e^{x}), t = e^{x}$
Then the original integral can be written as follows:
$\int \frac{1}{4 t + 6} d t$
$\int \frac{1}{4 x + 6} d x$
Calculate the table integral:
$\frac{1}{2} \int \frac{1}{2 x + 3} d x = \frac{\ln (2 x + 3)}{4}$
Answer:
$\ln ({(2 x + 3)}^{\frac{1}{4}}) + C$
To write down the final answer, it remains to substitute $\exp (x)$ instead of t.
$\frac{\ln (2 e^{x} + 3)}{4} + C$

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