Question

Evaluate the integral by using substitution

Answer 1

\[\int\limits_{?}^{?} x^3(x^4-10)^{45}dx \]
by making the substitution that u=x^4-10

Answer 2

ok u substituted u

now find what is dx in terms of du

Answer 3

\(u=x^4-10,\quad du=4x^3dx\)
\(\int\limits x^3(x^4-10)^{45}dx=\frac14\int u^{45}du\)

Answer 4

u=x^4-10
du/dx=4x^3
\[\int\limits \frac{ u ^{45} }{4 }dx=\frac{ 1 }{4 }\int\limits u ^{45}=\frac{ 1 }{4 }\times \frac{ u ^{46} }{ 46 }+c=\frac{ u ^{46} }{ 184 }+c\]

Answer 5

Don't forget to substitute (x^4 - 10) back in for you!

Answer 6

lol u*

Answer 7

okay okay.. I completely understand until you guys got to the parts after finding du.. can someone explain everything after that step by step?

Answer 8

\[\large \color{orangered}{u=x^4+10}\]Taking the derivative with respect to x,\[\large \frac{du}{dx}=4x^3\]Moving the dx to the other side gives us,\[\large du=4x^3dx\]Dividing both sides by 4 gives,\[\large \color{orangered}{\frac{1}{4}du=x^3dx}\]
Use the two orange pieces to make your substitution.

Answer 9

-10, sorry bout that. Shouldn't make much difference.

Answer 10

\[\huge \rightarrow \int\limits (u)^{45}\left(\frac{1}{4}du\right)\]