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Evaluate the indefinite integral: $\int(1+x/2)^{8}\,dx$

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Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate the indefinite integral:

\int(1+x/2)^{8}\,dx

Write Integral Format: First, let's write the integral in a proper format: $\int (1 + \frac{x}{2})^{8} \, dx$ .
Use Substitution: Now, let's use substitution. Let $u = 1 + \frac{x}{2}$ . Then, $du = \left(\frac{1}{2}\right) dx$ , or $dx = 2 du$ .
Substitute $u$ and $dx$ : Substitute $u$ and $dx$ in the integral: $\int u^8 \cdot 2 \, du$ .
Integrate $u^8$ : Now, integrate $u^8$ with respect to $u$ : rac{2 imes u^9}{9} + C.
Substitute back for u: Substitute back for u to get the function in terms of $x$ : $\frac{2 \cdot (1 + \frac{x}{2})^9}{9} + C$ .
Correct Integral: But wait, there's a mistake. The power of the function should be increased by $1$ , which means it should be $(1 + x/2)^9$ , not $(1 + x/2)^8$ . So the correct integral is $(2 \cdot (1 + x/2)^9)/9 + C$ .

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