2 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 2 years ago
Cezar Staff answered 2 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.