Evaluate the integral. 1. \displaystyle \int \frac { x ^ { 3 } } { e ^ { x ^ { 4 } } } d x 2. ...

Question:

Evaluate the integral.

{eq}\displaystyle \int \frac { x ^ { 3 } } { e ^ { x ^ { 4 } } } d x {/eq}

Integration by Substitution

In the method of substitution, it is possible to transform a difficult integral to an easier integral by using a substitution. The variable in the integrand will be expressed into a function of {eq}u {/eq}, hence this method is also called u-substitution, then the integral will be solved. Finally, the solution will be substituted back to its original variable.

Answer and Explanation:

Let {eq}\displaystyle I = \int \frac { x ^ { 3 } } { e ^ { x ^ { 4 } } } d x {/eq}


Use the substitution method,

Let {eq}\displaystyle u = x^4 \Rightarrow du = 4x^3\ dx \Rightarrow x^3\ dx = \frac {du}{4} {/eq}


Then,

{eq}\displaystyle \begin{align*} \Rightarrow I &= \int \frac {du}{4e^u} \\ \Rightarrow I &= \int \frac {e^{-u}du}{4} \\ \Rightarrow I &= \frac 14 \int e^{-u}du \\ \Rightarrow I &= \frac 14 \left ( \frac {e^{-u}}{-1} \right ) \\ \Rightarrow I &= \frac {-1}{4e^u} \\ \end{align*} {/eq}


Now substitute back {eq}u = x^4 {/eq} and add the constant of integration,

{eq}\displaystyle \displaystyle \int \frac { x ^ { 3 } } { e ^ { x ^ { 4 } } } d x = -\ \frac {1}{4e^{x^4}} + C {/eq}


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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