Chioma was asked to find $\int\left(18 x^{2}+3\right)\left(6 x^{3}+3 x\right)^{6} d x$ using $u$ substitution.

How should Chioma define $u$ ?

Question

Chioma was asked to find $\int\left(18 x^{2}+3\right)\left(6 x^{3}+3 x\right)^{6} d x$ using $u$ substitution.

How should Chioma define $u$ ?

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the inner function of the composite function to use for the $u$ substitution. ‖ ‖ step 2 ⋮ Define $u$ as the inner function, which is $u = 6 x^{3} + 3 x$. ‖ ‖ step 3 ⋮ Differentiate $u$ with respect to $x$ to find $\frac{du}{dx}$. ‖ ‖ step 4 ⋮ Calculate $\frac{du}{dx} = 18 x^{2} + 3$. ‖ ‖ step 5 ⋮ Notice that $\frac{du}{dx}$ matches the other part of the integrand, allowing us to substitute $du$ for $(18 x^{2} + 3)dx$. ‖ ‖ step 6 ⋮ Rewrite the integral in terms of $u$: $\int u^{6} du$. ‖ ‖ step 7 ⋮ Integrate $u^{6}$ with respect to $u$ to get $\frac{u^{7}}{7} + C$. ‖ ‖ step 8 ⋮ Substitute back the expression for $u$ to get the final answer in terms of $x$. ‖ ‖ step 9 ⋮ The final answer is $\frac{(6 x^{3} + 3 x)^{7}}{7} + C$. ‖ ∻Answer∻ ⚹ $\frac{(6 x^{3} + 3 x)^{7}}{7} + C$ ⚹ ∻Key Concept∻ ⚹$u$ substitution in integration⚹ ∻Explanation∻ ⚹In $u$ substitution, we identify an inner function within the integrand that, when differentiated, resembles another part of the integrand. This allows us to simplify the integral by substituting $du$ for that part of the integrand, making it easier to integrate.⚹◊What is the formula to find the derivative of $f$ given by $f^{\prime}(x)=\ln |x| \cdot x$?⍭ Generate me a similar question◊

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