Question

Math

Posted 3 months ago

```
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 $u$ ?
```

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Answer from Sia

Posted 3 months ago

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.

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