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Problem 984

# Integrate: $$\int\left(x^{2}+4\right)^{5} 2 x d x$$.

Expert verified
The integral of $$\int(x^2 + 4)^5 2x dx$$ can be computed using substitution method. Letting $$u = x^2 + 4$$, we find the integral to be $$\frac{1}{6}(x^2 + 4)^6 + C$$.
See the step by step solution

## Step 1: Choose a substitution

Let the substitution be: $$u = x^2 + 4$$.

## Step 2: Find the derivative of the substitution

Differentiate the substitution with respect to $$x$$: $$\frac{d u}{d x} = 2x$$.

## Step 3: Rewrite the integrand and find the differential

Rewrite the integrand using the substitution: $$\int (x^2 + 4)^5 (2x) dx$$ Now, find the differential in terms of $$u$$: $$d x = \frac{d u}{(2x)}$$

## Step 4: Substitute and simplify

Replace $$x$$, $$x^2+4$$, and $$dx$$ in the original integral: $$\int u^5 \frac{d u}{2x}$$ Notice that $$2x$$ from the fraction cancels out with the $$2x$$ from the integrand, and we are left with: $$\int u^5 d u$$

## Step 5: Integrate with respect to u

Now, integrate with respect to $$u$$: $$\int u^5 d u = \frac{1}{6} u^6 + C$$

## Step 6: Substitute back and simplify

Finally, replace $$u$$ with the original expression in terms of $$x$$: $$\frac{1}{6}(x^2 + 4)^6 + C$$ The final answer is: $\frac{1}{6}(x^2 + 4)^6 + C$

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