Answer:
∫[tex]6x^5(x^6-2)\,dx[/tex] = [tex]\frac{1}{2}(x^6-2)^2+C[/tex]
Step-by-step explanation:
To find:
∫[tex]6x^5(x^6-2)\,dx[/tex]
Solution:
Method of substitution:
Let [tex]x^6-2=t[/tex]
Differentiate both sides with respect to [tex]t[/tex]
[tex]6x^5\,dx=dt[/tex]
[use [tex](x^n)'=nx^{n-1}[/tex]]
So,
∫[tex]6x^5(x^6-2)\,dx[/tex] = ∫ [tex]t\,dt[/tex] = [tex]\frac{t^2}{2}+C_1[/tex] where [tex]C_1[/tex] is a variable.
(Use ∫[tex]t^n\,dt=\frac{t^{n+1} }{n+1}[/tex] )
Put [tex]t=x^6-2[/tex]
∫[tex]6x^5(x^6-2)\,dx[/tex] = [tex]\frac{1}{2}(x^6-2)^2+C_1[/tex]
Use [tex](a-b)^2=a^2+b^2-2ab[/tex]
So,
∫[tex]6x^5(x^6-2)\,dx[/tex] = [tex]\frac{1}{2}(x^6-2)^2+C_1=\frac{1}{2}(x^{12}+4-4x^6)+C_1=\frac{x^{12} }{2}-2x^6+2+C_1=\frac{x^{12} }{2}-2x^6+C[/tex]
where [tex]C=2+C_1[/tex]
Without using substitution:
∫[tex]6x^5(x^6-2)\,dx[/tex] = ∫[tex]6x^{11}-12x^5\,dx[/tex] = [tex]\frac{6x^{12} }{12}-\frac{12x^6}{6}+C=\frac{x^{12} }{2}-2x^6+C[/tex]
So, same answer is obtained in both the cases.
