Question

In Exercises 67 and \(68,\) make a substitution \(u=\cdots(\) an expression in \(x), \quad d u=\cdots .\) Then (a) integrate with respect to \(u\) from \(u(a)\) to \(u(b)\) . (b) find an antiderivative with respect to \(u,\) replace \(u\) by the expression in \(x,\) then evaluate from \(a\) to \(b\) . $$\int_{0}^{1} \frac{x^{3}}{\sqrt{x^{4}+9}} d x$$

Short answer

The value of the integral from 0 to 1 of \( \frac{x^{3}}{\sqrt{x^{4}+9}} dx \) is \( \frac{\sqrt{10}}{2} - \frac{3}{2} \).

Step-by-step solution

Identify the substitution

First, identify a part of the integrand that will simplify the integral once it's replaced with a single variable. In this case, the best choice for the substitution is \( u = x^4 + 9 \).

Calculate derivative of the substituted variable

Next, compute the derivative of \(u\) with respect to \(x\). The derivative \(du/dx\) is \( 4x^3 \). We also want to express \(dx\) in terms of \(du\), so we get \( dx = du / (4x^3) \).

Replace in the integral

Replace \(x^4 + 9\) with \(u\) and \(dx\) with \(du / (4x^3)\) in the original integral. The given integral then becomes: \( \int_{0}^{1} \frac{x^{3}}{4u^{1/2}} du \).

Simplify the Integral

The integral now simplifies to \( \int_{0}^{1} \frac{1}{4u^{1/2}} du \), since \(x^3\) in the numerator and \(x^3\) in the denominator cancel each other out.

Compute the integral

Now, compute the integral of \( \frac{1}{4u^{1/2}} \) which is \( \frac{u^{1/2}}{2} \). This gives the antiderivative of \(u\).

Replace \(u\) with the expression in \(x\)

Replace \(u\) by the expression in \(x\) and then evaluate it from \(0\) to \(1\). This turns out to be \( \frac{\sqrt{x^{4} + 9}}{2} \) evaluated from \(0\) to \(1\).

Evaluate the definite integral

Finally, compute \( \frac{\sqrt{x^{4} + 9}}{2} \bigg|_{0}^{1} \) which gives \( \frac{1}{2} \sqrt{10} - \frac{1}{2} \sqrt{9} = \frac{\sqrt{10}}{2} - \frac{3}{2} \).