How to Integrate ∫ a^x dx

In this video, we follow through the steps to solve the integral of a^x, where a is a positive real number.
We use the result ∫ e^x dx = e^x, and manipulate a^x to the form e^(xlna), then use a substitution.
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[@ryanrahuelvalentine2879]

Thank you so much sir. :)

[@MasterWuMathematics]

Most welcome!

[@meirabasinam4205]

👍👍👍

[@MasterWuMathematics]

👍👍👍

[@465HEY]

👍👍👍

[@samuelchung6383]

you're a fucking goat mate!! legend!

[@MasterWuMathematics]

Thanks mate! According to the Chinese zodiac, I’m actually a Pig!

[@muskansingh3196]

sir where did x go in differentiation please explain

[@vatsal512]

lna is constant therefore we have to differentiate (some constant × variable x ) we can do it by using power rule ...I hope you know differentiation of x ...thats what he did ..he basically differentiated x keeping constant aside

[@johnanthonygonzales1045]

sir what about if the exponent is not just x, say a^(8+3x)?

[@MohitPatel-ud1wm]

I think 3 will come down (as the coefficient of x in integration does) and rest according to formula

[@boysattitudes125]

@@MohitPatel-ud1wm yes you are right. First you can let ( x+3x) as t. Then expression will become a^t dt/3.

[@dalahmah]

What software (blackboard writing ) did you use ?

[@ShashankGautam-s6b]

What if the exponent is negative i.e. a^-x?

[@MasterWuMathematics]

Just make the substitution u = -x and try that.

[@ShashankGautam-s6b]

@@MasterWuMathematics okay , thanks 👍

[@lancethederp]

what does a represent?

[@bharatsachdeva6683]

some constant

[@muhammadakbar6070]

what if a is a negative number, since you cant log that

[@MasterWuMathematics]

Try graphing for example (-2)^x. You'll quickly find that this is not a function.

[@muhammadakbar6070]

@@MasterWuMathematics oh right, it's just a bunch of points, thanks

[@MasterWuMathematics]

Correct, at least it's not a function in terms of real numbers.

[@ZahraaAli-yt6dk]

Perfect 🤩

[@rinkihaldar3542]

Op