Question

Use the integral \(\int\left(x^{2}+a^{2}\right)^{-1 / 2} d x\) to prove that \(\int \frac{d x}{\left(x^{2}+a^{2}\right)^{3 / 2}}=\frac{x}{a^{2}\left(x^{2}+a^{2}\right)^{1 / 2}}+C\)

Short answer

Question: Show that \(\int \frac{dx}{\left(x^{2}+a^{2}\right)^{3 / 2}}\) can be integrated to obtain \(\frac{x}{a^{2}\sqrt{x^{2}+a^{2}}}+C\). Solution: We started by differentiating the given integral \(\int\left(x^{2}+a^{2}\right)^{-1 / 2}d x\) to obtain \(\frac{d}{dx} \int\left(x^{2}+a^{2}\right)^{-1 / 2} dx = x\left(x^{2}+a^{2}\right)^{-3 / 2}\). Then, integrating both sides, we reached the expression \(\int \frac{dx}{\left(x^{2}+a^{2}\right)^{3 /2}}= \frac{x}{a^{2}\left(x^{2}+a^{2}\right)^{1 / 2}} + C\), which is the desired proof.

Step-by-step solution

Differentiate the given integral

We are given the integral \(\int\left(x^{2}+a^{2}\right)^{-1 / 2}d x\). We need to find the derivative of this integral with respect to \(x\). Let \(F(x)=\int\left(x^{2}+a^{2}\right)^{-1 / 2}d x\). By the first fundamental theorem of calculus, we have \(F'(x)=\frac{dF(x)}{dx}\). Differentiating both sides of \(F(x)\), we'll get the derivative: $$ \frac{d}{dx} \int\left(x^{2}+a^{2}\right)^{-1 / 2} dx = \left(x^{2}+a^{2}\right)^{-1 / 2} \frac{d}{dx}\left(x^{2}+a^{2}\right) $$

Calculate the derivative

Now we need to find the derivative of the expression inside the integral, which is given by: $$ \frac{d}{dx}\left(x^{2}+a^{2}\right) = 2x $$

Plug in the derivative of the expression into the formula

Plug in the value of the derivative we just calculated into the formula from Step 1: $$ \frac{d}{dx} \int\left(x^{2}+a^{2}\right)^{-1 / 2} dx = 2x\left(x^{2}+a^{2}\right)^{-1 / 2} $$

Simplify the derivative

Multiply the expression by \(\frac{1}{2}\) to simplify the derivative: $$ \frac{d}{dx} \int\left(x^{2}+a^{2}\right)^{-1 / 2} dx = x\left(x^{2}+a^{2}\right)^{-3 / 2} $$ This gives us: $$ \int \frac{dx}{\left(x^{2}+a^{2}\right)^{3 /2}}=x\left(x^{2}+a^{2}\right)^{-3/2} $$

Integrate the expression

Now, we can integrate the expression to get the result: $$ \int \frac{dx}{\left(x^{2}+a^{2}\right)^{3 /2}}= \int x\left(x^{2}+a^{2}\right)^{-3/2}dx $$ Based on the given integral, we know that: $$ \int \left(x^{2}+a^{2}\right)^{-1 / 2} dx = \frac{x}{a^{2}\sqrt{x^{2}+a^{2}}}+C $$ Taking the derivative of this expression with respect to \(x\), we get the desired proof: $$ \int \frac{dx}{\left(x^{2}+a^{2}\right)^{3 /2}}= \frac{x}{a^{2}\left(x^{2}+a^{2}\right)^{1 / 2}} + C $$