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Chapter 1: Problem 6

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

Expert verified
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

01

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: $$ \frac{d}{dx} \int\left(x^{2}+a^{2}\right)^{-1 / 2} dx = \left(x^{2}+a^{2}\right)^{-1 / 2} \frac{d}{dx}\left(x^{2}+a^{2}\right) $$
02

Calculate the derivative

Now we need to find the derivative of the expression inside the integral, which is given by: $$ \frac{d}{dx}\left(x^{2}+a^{2}\right) = 2x $$
03

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: $$ \frac{d}{dx} \int\left(x^{2}+a^{2}\right)^{-1 / 2} dx = 2x\left(x^{2}+a^{2}\right)^{-1 / 2} $$
04

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: $$ \int \frac{dx}{\left(x^{2}+a^{2}\right)^{3 /2}}=x\left(x^{2}+a^{2}\right)^{-3/2} $$
05

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: $$ \int \frac{dx}{\left(x^{2}+a^{2}\right)^{3 /2}}= \frac{x}{a^{2}\left(x^{2}+a^{2}\right)^{1 / 2}} + C $$

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Most popular questions from this chapter

Evaluate the following integrals: (i) \(\int \ln \left(x+\sqrt{x^{2}+a^{2}}\right) d x\) (ii) \(\int \ln ^{2}\left(x+\sqrt{1+x^{2}}\right) d x\) (iii) \(\int x^{2} \ln \frac{1+x}{1-x} d x\) (iv) \(\int \frac{\ln x}{(x-1)^{3}} d x\)

Find a substitution to reduce the integral \(\int \mathrm{R}(\mathrm{x}, \mathrm{y}) \mathrm{dx}\) when \(\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{2}=\mathrm{a}^{2}\left(\mathrm{x}^{2}-\mathrm{y}^{2}\right)\)

Evaluate the following integrals: (i) \(\int x \sin x \cos ^{2} x d x\) (ii) \(\int x \sec ^{2} x \tan x d x\) (iii) \(\int x \cos x \cos 2 x d x\)

Evaluate the following integrals: (i) \(\int \frac{2 x^{3}+3 x^{2}+4 x+5}{2 x+1} d x\) (ii) \(\int\left(\frac{x^{-6}-64}{4+2 x^{-1}+x^{-2}}, \frac{x^{2}}{4-4 x^{-1}+x^{-2}} \frac{4 x^{2}(2 x+1)}{1-2 x}\right) \mathrm{dx}\) (iii) \(\int\left(\frac{\sqrt{x}}{2}-\frac{1}{2 \sqrt{x}}\right)^{2}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}-\frac{\sqrt{x}+1}{\sqrt{x}-1}\right) d x\) (iv) \(\int \frac{\sqrt{1-x^{2}}+1}{\sqrt{1-x}+1 / \sqrt{1+x}} d x\).

Evaluate the following integrals: (i) \(\int \sin (\ln x) \mathrm{d} x\) (ii) \(\int \mathrm{e}^{x} \sin x \sin 3 x d x\) (iii) \(\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x\) (iv) \(\int x^{3} \tan ^{-1} x d x\)
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