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Chapter 54: Chapter 54

Problem 984

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Integrate: \(\int\left(x^{2}+4\right)^{5} 2 x d x\).

Short Answer

Expert verified
The integral of \(\int(x^2 + 4)^5 2x dx\) can be computed using substitution method. Letting \(u = x^2 + 4\), we find the integral to be \(\frac{1}{6}(x^2 + 4)^6 + C\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Choose a substitution

Let the substitution be: \(u = x^2 + 4\).

Step 2: Find the derivative of the substitution

Differentiate the substitution with respect to \(x\): \(\frac{d u}{d x} = 2x\).

Step 3: Rewrite the integrand and find the differential

Rewrite the integrand using the substitution: \(\int (x^2 + 4)^5 (2x) dx\) Now, find the differential in terms of \(u\): \(d x = \frac{d u}{(2x)}\)

Step 4: Substitute and simplify

Replace \(x\), \(x^2+4\), and \(dx\) in the original integral: \(\int u^5 \frac{d u}{2x}\) Notice that \(2x\) from the fraction cancels out with the \(2x\) from the integrand, and we are left with: \(\int u^5 d u\)

Step 5: Integrate with respect to u

Now, integrate with respect to \(u\): \(\int u^5 d u = \frac{1}{6} u^6 + C\)

Step 6: Substitute back and simplify

Finally, replace \(u\) with the original expression in terms of \(x\): \(\frac{1}{6}(x^2 + 4)^6 + C\) The final answer is: \[\frac{1}{6}(x^2 + 4)^6 + C\]

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