$$ 1 \leftarrow \quad \begin{array}{l} ext { Determine the following indefinite integral. } \ \int(4 s+7)^{2} d s\end{array} $$

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$$ 1 \leftarrow \quad \begin{array}{l}	ext { Determine the following indefinite integral. } \ \int(4 s+7)^{2} d s\end{array} $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int(4s+7)^{2} ds $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \left(4s+7ight)^{2} ds$$
- step1: Use the substitution $$ds=\frac{1}{4} dt$$ to transform the integral$$:$$
  $$\int \left(4s+7ight)^{2}	imes \frac{1}{4} dt$$
- step2: Simplify:
  $$\int \frac{\left(4s+7ight)^{2}}{4} dt$$
- step3: Use the substitution $$t=4s+7$$ to transform the integral$$:$$
  $$\int \frac{t^{2}}{4} dt$$
- step4: Rewrite the expression:
  $$\int \frac{1}{4}t^{2} dt$$
- step5: Use properties of integrals:
  $$\frac{1}{4}	imes \int t^{2} dt$$
- step6: Evaluate the integral:
  $$\frac{1}{4}	imes \frac{t^{2+1}}{2+1}$$
- step7: Simplify:
  $$\frac{1}{4}	imes \frac{t^{3}}{3}$$
- step8: Multiply the terms:
  $$\frac{t^{3}}{4	imes 3}$$
- step9: Multiply the terms:
  $$\frac{t^{3}}{12}$$
- step10: Substitute back:
  $$\frac{\left(4s+7ight)^{3}}{12}$$
- step11: Add the constant of integral C:
  $$\frac{\left(4s+7ight)^{3}}{12} + C, C \in \mathbb{R}$$
The indefinite integral of $$(4s+7)^{2}$$ with respect to $$s$$ is $$\frac{(4s+7)^{3}}{12} + C$$, where $$C$$ is an arbitrary constant.
The integral of $$(4s+7)^{2}$$ with respect to $$s$$ is $$\frac{(4s+7)^{3}}{12} + C$$.

\( 1 \leftarrow \quad \begin{array}{l}\text { Determine the following indefinite integral. } \\ \int(4 s+7)^{2} d s\end{array} \)

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