Question

Integrate the integral (step-wise) -
$(5x^{2} - 8x + 5)dx$


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Answer 1

Given :

$\bigstar\:\underline{\boxed{\bf{\dfrac{dy}{dx}=5x^2-8x+5}}}$

To Find :

➳ We have to integrate the integral.

Remember :

$:\implies\sf\:\dfrac{dp}{dq}=q^n+1$

$:\implies\sf\:p=\int(dp)=\int(q^n+1)\:dq$

$:\implies\bf\:p=\dfrac{q^{n+1}}{(n+1)}+q$

SoluTion :

$\leadsto\tt\:\dfrac{dy}{dx}=5x^2-8x+5$

$\leadsto\tt\:dy=(5x^2-8x+5)\:dx$

$\leadsto\tt\:y=\int(dy)=\int(5x^2-8x+5)\:dx$

$\leadsto\tt\:y=\dfrac{5x^{2+1}}{(2+1)}-\dfrac{8x^{1+1}}{(1+1)}+5x$

$\leadsto\tt\:y=\dfrac{5x^3}{3}-\dfrac{8x^2}{2}+5x$

$\leadsto\underline{\boxed{\bf{y=\dfrac{5x^3}{3}-4x^2+5x}}}$

Answer 2

Given ,

The function is y = 5(x)² - 8x + 5

Integratraing y wrt x , we get

$\rightarrow \sf \int{5 {(x)}^{2} - 8x + 5 } \: \: dx \\ \\ \rightarrow \sf 5\int{ {(x)}^{2}} \: dx- 8\int{x} \: dx+ \int{5} \: dx \\ \\ \sf \rightarrow 5 \{ \frac{ {(x)}^{2 + 1} }{2 + 1} \} - 8 \{ \frac{ {(x)}^{1 + 1} }{1 + 1} \} + 5x\\ \\ \sf \rightarrow 5 \{ \frac{ {(x)}^{3} }{3} \} - 8 \{ \frac{ {(x)}^{2} }{2} \} + 5x \\ \\ \sf \rightarrow 5 \{ \frac{ {(x)}^{3} }{3} \} - 4 {(x)}^{2} + 5x + c$

Remmember :

$\sf \mapsto \int{ {(x)}^{n} } \: dx = \frac{ {(x)}^{n + 1} }{n + 1} \\ \\ \sf \mapsto \int{ (k) } \: dx = kx$