# Carrying out a substituting to evaluate $\int (x + 1) (x^2 + 2 x)^5dx$

The problem is: $$\int { (x+1)({ x }^{ 2 } } +2x{ ) }^{ 5 }dx$$

The next step given by WolframAlpha is $$\int { (x+1)({ x }^{ 2 } } +2x{ ) }^{ 5 }dx\\ =\quad \frac { 1 }{ 2 } \int { { u }^{ 5 }du }$$

(While I realize I am doing somthing worng)

The steps I am taking are : $\int { (x+1)({ x }^{ 2 } } +2x{ ) }^{ 5 }dx$.

Let $u={ x }^{ 2 }+2x, \, du = 2x+2\, dx$. Then I see I can rewrite $du$ as $du = 2(x+1)dx$ giving me $\frac { du }{ 2 } = (x+1)\, dx$.

I can now see where the $\frac { 1 }{ 2 }$ is coming from, but I cannot seem to visualize the next steps to get to

$$\int { (x+1)({ x }^{ 2 } } +2x{ ) }^{ 5 }dx=\frac { 1 }{ 2 } \int { { u }^{ 5 }du }$$

• You are basically done. You just need to substitute $x^2+2x=u$ and $(x+1)dx=\frac12du$. May 4, 2016 at 6:04
• What happens when you replace $(x^2+2x)^5$ with $u^5$ and $(x+1) \mathrm{d}x$ with $\dfrac{1}{2} \mathrm{d}u$? May 4, 2016 at 6:05
• Please don't put large pieces of text in LaTeX. Also, you use the curly braces unnecessarily many times. May 4, 2016 at 6:12

You already state in your question that you found that $$\frac{\mathrm{d}u}{2} = (x+1)\mathrm{d}x$$
when doing the substitution $u=x^2+2x$. When doing this substitutiuon on the integral, we get $$\int (x+1)(x^2+2x)^5 \mathrm{d}x = \int (x+1)u^5 \mathrm{d}x = \int u^5 (x+1)\mathrm{d}x = \int u^5 \frac{\mathrm{d}u}{2} = \frac{1}{2} \int u^5 \mathrm{d}u$$
Since $(x^2 + 2 x)^5 = u^5$, we can use the equation $(x + 1) \,dx = \frac{1}{2} du$ to write the integral as $$\int \underbrace{(x^2 + 2 x)^5}_{u^5} \,\underbrace{(x + 1) \,dx}_{\frac{1}{2} du} = \int u^5 \cdot \frac{1}{2} du = \frac{1}{2} \int u^5 \,du .$$