Step by step solution to "Evaluate the integrals. ∫x3ex4dx"

Secondary
Calculus and Analysis
Calculus 2
Applications of integrals

kuvitia9f Answered2021-12-31

Evaluate the integrals.

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

Answer & Explanation

ambarakaq8

Expert

2022-01-01Added 31 answers

Step 1
Solution -
Given integral -

y = \int x^{3} e^{x^{4}} d x

Let,

t = x^{4}

differentiating on both sides w.r.t x,

\frac{d t}{d x} = 4 x^{3}

d t = 4 x^{3} d x

\frac{d t}{4} = x^{3} d x

Step 2
Now substituting these values in the given integral,

y = \frac{1}{4} \int e^{t} d t

y = \frac{1}{4} [e^{t}] + C

where C is the constant.

Philip Williams

Expert

2022-01-02Added 39 answers

Step 1
This problem can be solved using a u-substitution. Let $u = x^{4}$ . Then $d u = 4 x^{3} d x$ .
$\int x^{3} e^{x^{4}} d x = \int \frac{e^{u}}{4} d u = \frac{e^{u}}{4} + c = \frac{e^{x^{4}}}{4} + c$
Result:
$\frac{e^{x^{4}}}{4} + c$

Vasquez

Expert

2022-01-07Added 457 answers

Step 1
Substitute P $x^{4} = u and 4 x^{3} d x = d u \Rightarrow x^{3} d x = \frac{1}{4} d u$
$\int x^{3} e^{x^{4}} d x = \frac{1}{4} \int e^{u} d u = \frac{1}{4} e^{u} + C$
Substitute back $u = x^{4}$
$= \frac{1}{4} e^{x^{4}} + C$
Result:
$\frac{1}{4} e^{x^{4}} + C$