$$ \int \frac{ ext { Integration using u-substitution }}{6(x+1)(x+2)^{7} d x} $$

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$$ \int \frac{	ext { Integration using u-substitution }}{6(x+1)(x+2)^{7} d x} $$

UpStudy AI · Accepted Answer

To solve the given integral using the u-substitution method, we can substitute $$ u = x + 2 $$ and then find the derivative of $$ u $$ with respect to $$ x $$ to substitute it into the integral.

Given integral: $$ \int \frac{1}{6(x+1)(x+2)^{7}} dx $$

Let $$ u = x + 2 $$, then $$ du = dx $$.

Substitute $$ u $$ and $$ du $$ into the integral:

$$ \int \frac{1}{6(x+1)(x+2)^{7}} dx = \int \frac{1}{6(u-1)u^{7}} du $$

Now, we can solve this integral using the u-substitution method.
To solve the integral $$ \int \frac{1}{6(x+1)(x+2)^{7}} dx $$ using u-substitution, let $$ u = x + 2 $$, so $$ du = dx $$. Substituting, the integral becomes $$ \int \frac{1}{6(u-1)u^{7}} du $$.

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