Evaluate the integral by making an appropriate substitution. \int\frac{x}{x^{2} + 4} dx

Question:

Evaluate the integral by making an appropriate substitution.

{eq}\displaystyle\; \int\frac{x}{x^{2} + 4}\,dx {/eq}

Indefinite Integrals:

The substitution of the variable is applied with the indefinite integral will give the final result with the same variable after we have done the back substitution of the intermediate integral result.

Answer and Explanation:


The indefinite integral given here is;

{eq}\int\frac{x}{x^{2} + 4}\,dx {/eq}

So here we will apply the proper substitution of:

{eq}u=x^2+4\\ {/eq}

and the take the derivative as follows:

{eq}\frac{du}{dx}=2x\\ {/eq}

So now the integral will be substituted to:

{eq}\int\frac{x}{x^{2} + 4}\,dx=\int \frac{1}{2u}du\\ =\frac{1}{2}\ln \left|u\right|+c\\ =\frac{1}{2}\ln \left|x^2+4\right|+C\\ {/eq}

is the result of integral after the back substitution.


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Indefinite Integrals as Anti Derivatives

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