Integration by parts vs integration by parts with substitution different answers

Question

I'm trying to integrate

$$\int(5x^4+1)\ln(x^5+x)dx$$

When I use an online calculator (it first uses substitution and then integration by parts) I get the answer

$$(x^5+x)\ln(x^5+x)-x^5-x$$

but when I try a different approach (without substitution only with integration by parts) I get

$$(x^5+x)\ln(x^5+x)-x$$

(I change the variable of $dx$ to $x^5+x$)

Answer 1

It seem's that you forgot something since diferentiation of $(x^5+x)\ln(x^5+x)-x$ gives: \begin{align*} D_x\left[(x^5+x)\ln(x^5+x)-x\right]&=(5x^4+1)\ln(x^5+x)+5x^4+1-1\\ &=(5x^4+1)\ln(x^5+x)+5x^4 \end{align*} And in the expression $$(x^5+x)\ln(x^5+x)-x$$ the last $x$ must be $x^5+x$ because the change that you used in order to integrate, i.e. $x^5+x$ instead of $x$.

Answer 2

Since your integral looks like $\int g’\ln(g)$ you just need to find a primitive of $\ln( x)$, namely $x\cdot\ln(x)-x$, and just plug in $x^5+x$ for $g$.

Answer 3

Integration with x only $$ \begin{aligned} \int\left(5 x^4+1\right) \ln \left(x^5+x\right) d x&=\int \ln \left(x^5+x\right) d\left(x^5+x\right) \\ &=\left(x^5+x\right) \ln \left(x^5+x\right)-\int\left(x^5+x\right) \frac{5 x^4+1}{x^5+x} d x \\ &=\left(x^5+x\right) \ln \left(x^5+x\right)-\int\left(5 x^4+1\right) d x \\ &=\left(x^5+x\right) \ln \left(x^5+x\right)-x^5-x+C \end{aligned} $$ Integration with a new variable u

Let $u=x^5+x$, then $d u=\left(5 x^4+1\right) d x$ $$ \begin{aligned} \int\left(5 x^4+1\right) \ln \left(x^5+x\right) d x &=\int \ln u d u \\ &=u \ln u-\int u \cdot \frac{1}{u} d u \\ &=u \ln u-u+C \\ &=\left(x^5+x\right) \ln \left(x^5+x\right)-x^5-x+C \end{aligned} $$