\[\int \sqrt{7x+9} \, dx\]

1
Use Power Substitution.
Let u=7x+9u=\sqrt{7x+9}, x=u297x=\frac{{u}^{2}-9}{7}, and dx=2u7dudx=\frac{2u}{7} \, du

2
Substitute variables from above.
u×2u7du\int u\times \frac{2u}{7} \, du

3
Use Constant Factor Rule: cf(x)dx=cf(x)dx\int cf(x) \, dx=c\int f(x) \, dx.
27u2du\frac{2}{7}\int {u}^{2} \, du

4
Use Power Rule: xndx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
2u321\frac{2{u}^{3}}{21}

5
Substitute u=7x+9u=\sqrt{7x+9} back into the original integral.
27x+9321\frac{2{\sqrt{7x+9}}^{3}}{21}

6
Add constant.
2(7x+9)3221+C\frac{2{(7x+9)}^{\frac{3}{2}}}{21}+C

Done

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