Calculus and AnalysisFind the general indefinite integrals
3 years ago
Find the general indefinite integrals integral of x^(2) square root of (x^(3)+1)dx integral of pi sin x cos^(3)x dx
1 Answers
Best Answer
Cezar Staff answered 3 years ago
Step 1 Given: I=int x^(2)sqrt(x^(3)+1)dx for evaluating given integral, in given integral we substitute x^(3)+1=t...(1) now, differentiating equation (1) with respect to x (d)/(dx)(x^(3)+1)=(dt)/(dx) (because (d)/(dx)(x^(n))=nx^(n-1)) 3x^(2)+0=(dt)/(dx) 3x^(2)=(dt)/(dx) x^(2)dx=(dt)/(3) Step 2 now, replace (x^(3)+1) with t, x^(2)dx with (dt)/(3) in given integral so, int x^(2)sqrt(x^(3)+1)dx=(1)/(3)int sqrt(t)dt =(1)/(3)((t^((1)/(2)+1))/((1)/(2)+1))+c =(1)/(3)((t^((3)/(2)))/(((3)/(2))))+c =(2t^((3)/(2)))/(9)+c =(2(x^(3)+1)^((3)/(2)))/(9)+c hence, given integral is equal to (2(x^(3)+1)^((3)/(2)))/(9)+c.
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