Answer to Question #136204 in Calculus for qwerty

Question #136204
\int \left(x^3+1\right)^5x^2dx
1
Expert's answer
2020-10-05T16:48:24-0400

(x3+1)5x2dx\int \left(x^3+1\right)^5x^2dx


Here, we apply substitution method I.e.


Let u=x3+1u = x^3 + 1


    dudx=3x2\implies \frac{du}{dx} = 3x^2


    dx=13x2du\implies dx= \frac{1}{3x^2}du


Replacing back, we have;


(u)5x213x2du\int (u)^5 \cancel{x^2} \frac{1}{3 \cancel{x^2} }du =13u5du= \frac{1}{3} \int u^5 du


Now we solve;


u5du\int u^5 du


Here, we apply power rule;


Where we let;


undu=un+1n+1\int u^n du = \frac{u^{n+1}}{n+1} With n=5n = 5 , we have;


=u66= \frac{u^6}{6}


Replacing this back to 13u5du\frac{1}{3} \int u^5 du we have;


u618\frac{u^6}{18}


Now we undo the substitution

u=x3+1u = x^3 +1


(x3+1)618\frac{(x^3 +1)^6}{18} and we add a C to this, hence the final answer i.e.


(x3+1)618+C\frac{(x^3 +1)^6}{18} + C


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