Question

What is the integration of 10x^4(2x^5-4)

Answer: (2x^5-4)^2/2

Answer 1

Answer:

let 2x^5-4=t

differentiate it

then,10x^4dx=dt

10x^4*t dx=tdt

t^2/2

(2x^5-4)^2/2

Answer 2

Answer:

$\displaystyle \longrightarrow \frac{\left(2x^5-4\right)^2}{2}$

Explanation:

Let :

$\displaystyle \text{I}=10x^4\left(2x^5-4\right)$

$\displaystyle \text{I}=\int {10x^4\left(2x^5-4\right)} \, dx$

Let :

$\displaystyle u=2x^5-4$

Diff. w.r.t. x :

$\displaystyle \frac{du}{dx} =\left(2x^5-4\right)'$

$\displaystyle \longrightarrow \frac{du}{dx} =10x^4$

$\displaystyle \longrightarrow dx =\frac{du}{10x^4}$

$\displaystyle \text{I}=\int {10x^4\left(2x^5-4\right)} \, dx$

$\displaystyle \longrightarrow \text{I}=\int{10x^4(u)} \, \frac{du}{10x^4}$

$\displaystyle \longrightarrow \text{I}=\int {(u)} \, du$

$\displaystyle \longrightarrow \text{I}=\frac{u^2}{2}$

$\displaystyle \longrightarrow \text{I}=\frac{\left(2x^5-4\right)^2}{2}$

Hence we get required answer!