Find the indefinite integral. Please show work.

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Find the indefinite integral. Please show work.

Mathematics
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\[\int\limits_{?}^{?}(x+1)5^(x+1)^2\]
that (x+1)^2 is an exponent
\[ \int(x+1)5^{(x+1)^2}dx \]That?

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yes
\[\Huge \int\limits_{}^{}(x+1)(5)^{(x+1)^2}\]
substitution u==x+1 du=dx \[\text{}=\int\limits 5^{u^2} u \, du\] substitution again t=u^2 dt=2udu \[\frac{1}{2}\int\limits 5^t \, dt\] \[\frac{5^t}{2 \ln (5)}+c\] \[\frac{5^{(x+1)^2}}{\log (25)}+c\]
ln=log
Okay, so u=x+1 and we have \[ \int u\cdot 5^{u^2}du=\int u\cdot e^{u^2\log5}du=\frac{1}{2\log 5}\int 2\log5u\cdot e^{u^2\log5}du=\frac{e^{u^2\log5}}{2\log 5}=\frac{5^{(x+1)^2}}{2\log5} \]
Oh, Sam's way is a little easier.
well the answer the book gives me looks like
\[1/2(5^{(x+1)^2}/\ln5)+C\]
Yep, that's the exact same thing as my answer and Sam's answer, just written slightly different.
i just dont understand how they get to that
Which part of the method Sam and I used did you not follow? (Sam's is a bit easier to follow because he does substitution twice and I only used it once)
i understand u substitution
5^u^2 = (ln5)u^2 right?
\[ 5^{u^2}=e^{(\log5)u^2} \]
\[ \int a^xdx=\int e^{x\log a}dx=\frac{1}{\log a}\int(\log a) e^{x\log a}dx=\frac{e^{x\log a}}{\log a}=\frac{a^x}{\log a} \\\int a^xdx=\frac{a^x}{\log a} \]
ok here is where im confused
how does 5^t become 5^t/ln5
in sams answer
He integrates it, using the formula I derived in my last comment.
well thats where im lost. I do not know how to integrate that
\[1/2\int\limits5^t dt\]
i would say that that is (ln5)t
and obviously im wrong lol
I just showed you how to integrate that in my comment, did you not see that? For \(a\) as any constant.
\[\int a^xdx=\int e^{x\log a}dx=\frac{1}{\log a}\int(\log a) e^{x\log a}dx=\frac{e^{x\log a}}{\log a}=\frac{a^x}{\log a} \\\int a^xdx=\frac{a^x}{\log a}\]
k ill write that down and try to wrap my brain around it
some things are just beyond me i guess
You just have to remember that \(e^{x^y}=e^{xy}\), which makes it so that \(a^x=(e^{\log a})^x=e^{x\log a}\)
Sorry, that first part should read \((e^x)^y=e^{xy}\)
luckily i have a 99.8 average in this class so if i miss this on the final it shouldnt hurt much >.<
Just revisit the rules for differentiating/integrating exponential and logarithmic functions. They come in handy for a lot of tricky integrals.
i try, i think my brain is overloaded. It is finals week

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