Solved: "Use the Change of Variables Formula to evaluate..."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

burkinaval1b

Answered question

2022-01-03

Use the Change of Variables Formula to evaluate the definite integral.

\int_{0}^{4} x \sqrt{x^{2} + 9} d x

Answer & Explanation

Foreckije

Beginner2022-01-04Added 32 answers

Step 1
Using Change of variables we have to evaluate the definite integral.
$\int_{0}^{4} x \sqrt{x^{2} + 9} d x$
Step 2
$\int_{0}^{4} x \sqrt{x^{2} + 9} d x$
Let us substitute $x^{2} + 9$ by t
So, when x=0, t=9
and when x=4, t=25
Differentiating $x^{2} + 9 = t \Rightarrow 2 x d x = d t \Rightarrow x d x = \frac{d t}{2}$
So we can write the integral $\int_{0}^{4} x \sqrt{x^{2} + 9} d x$ as $\frac{1}{2} \int_{9}^{25} \sqrt{t} d t$
Now $\frac{1}{2} \int_{9}^{25} \sqrt{t} d t = \frac{t^{\frac{3}{2}}}{3} ∣_{9}^{25} = \frac{125}{3} - \frac{27}{3} = \frac{98}{3}$
So, $\int_{0}^{4} x \sqrt{x^{2} + 9} d x = \frac{98}{3}$

Corgnatiui

Beginner2022-01-05Added 35 answers

\int_{0}^{4} x \sqrt{x^{2} + 9} d x

Let us put the expression 2 * x under the differential sign, i.e.:

2 x d x = d (x^{2}), t = x^{2}

Then the original integral can be written as follows:

\int \frac{\sqrt{t + 9}}{2} d t

We make a change of variables:
t=x+9
Then:

\int \frac{\sqrt{x + 9}}{2} d x = \int \frac{\sqrt{x + 9}}{2} \cdot d (x + 9) = \int \frac{\sqrt{t}}{2} d t = \frac{t^{\frac{3}{2}}}{3}

Back to x:

\frac{{(x + 9)}^{\frac{3}{2}}}{3}

Answer:

\frac{{(x + 9)}^{\frac{3}{2}}}{3} + C

To write the final answer, it remains to substitute

x^{2}

instead of t.

\frac{{(x^{2} + 9)}^{\frac{3}{2}}}{3} + C

Lets

karton

Expert2022-01-11Added 613 answers

$Step 1 Given \int_{0}^{4} x \sqrt{x^{2} + 9} d x \int x \sqrt{x^{2} + 9} d x \int \frac{1}{2} \times \sqrt{t} d t \frac{1}{2} \times \int \sqrt{t} d t Transform the expression \frac{1}{2} \times \int t^{\frac{1}{2}} d t Step 2 Evaluate the integral \frac{1}{2} \times \frac{2 t \sqrt{t}}{3} \frac{1}{2} \times \frac{2 (x^{2} + 9) \sqrt{x^{2} + 9}}{3} \frac{(x^{2} + 9) \sqrt{x^{2} + 9}}{3} \frac{(x^{2} + 9) \sqrt{x^{2} + 9}}{3} |_{0}^{4} Calculate the expression \frac{(4^{2} + 9) \sqrt{4^{2} + 9}}{3} - \frac{(0^{2} + 9) \sqrt{0^{2} + 9}}{3} Answer: \frac{98}{3}$

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