Copyright

Determine the following indefinite integral.

{eq}\displaystyle \int 7 x e^{\displaystyle 3x^2 + 2}\ dx {/eq}

Question:

Determine the following indefinite integral.

{eq}\displaystyle \int 7 x e^{\displaystyle 3x^2 + 2}\ dx {/eq}

Integration:


Calculating an integral is called integration. Mathematicians utilise integrals to determine a variety of useful quantities, including areas, volumes, displacement, etc. For antiderivatives, indefinite integrals are utilised. When we discuss integrals, we often refer to definite integrals.

Answer and Explanation: 1


Given:


  • We have given the function as {eq}7x{{e}^{3{{x}^{2}}+2}}{/eq}.


  • The objective is to find the integration of given function.


Now,


{eq}\int{7x{{e}^{3{{x}^{2}}+2}}dx}=7\int{x{{e}^{3{{x}^{2}}+2}}dx}{/eq}


Suppose,


{eq}\begin{align*} {{x}^{2}}&=u \\ 2xdx&=du \end{align*}{/eq}


Now, substitute this value in given integral,


{eq}\begin{align*} 7\int{x{{e}^{3{{x}^{2}}+2}}dx}&=7\int{{{e}^{3u+2}}\frac{du}{2}} \\ & =\frac{7}{2}\int{{{e}^{3u+2}}du} \end{align*}{/eq}


Now,


{eq}\begin{align*} \frac{7}{2}\int{{{e}^{3u+2}}du}&=\frac{7}{2}\left( \frac{{{e}^{3u+2}}}{3} \right)+C \\ & =\frac{7}{6}{{e}^{3u+2}}+C \\ & =\frac{7}{6}{{e}^{3{{x}^{2}}+2}}+C \end{align*}{/eq}


Hence, {eq}\int{7x{{e}^{3{{x}^{2}}+2}}dx}=\frac{7}{6}{{e}^{3{{x}^{2}}+2}}+C{/eq}.


Learn more about this topic:

Loading...
Integration Problems in Calculus: Solutions & Examples

from

Chapter 13 / Lesson 13
26K

Learn what integration problems are. Discover how to find integration sums and how to solve integral calculus problems using calculus example problems.


Related to this Question

Explore our homework questions and answers library