Find the anti derivative of
u^-2(1-u^2+u^4)du. Need steps so I can try to grasp this. Thanks. I couldn't figure out how to write the anti deriv. symbol but it is in front on the u^-2.
Thanks
To find the antiderivative of the given expression, u^-2(1-u^2+u^4)du, we can use the following steps:
Step 1: Distribute u^-2 to each term:
u^-2(1-u^2+u^4)du = u^-2(1/u^2 - u^2/u^2 + u^4/u^2)du
Simplifying this expression gives us:
1/u^2 - 1 + u^2
Step 2: Find the antiderivative of each term separately.
The antiderivative of 1/u^2 can be found using the power rule of integration, which states that ∫(x^n)dx = (x^(n+1))/(n+1) + C. Applying this rule, we have:
∫(1/u^2)du = ∫u^-2du
= u^(-2+1)/(-2+1) + C
= u^(-1)/(-1) + C
= -1/u + C1
The antiderivative of -1 can be found by treating it as a constant, so we have:
∫(-1)du = -u + C2
The antiderivative of u^2 can be found using the power rule once again:
∫(u^2)du = (u^(2+1))/(2+1) + C
= u^3/3 + C3
Therefore, the antiderivative of the expression 1/u^2 - 1 + u^2 is:
∫(1/u^2 - 1 + u^2)du = -1/u + u^3/3 - u + C
Combining like terms, we have:
∫(1/u^2 - 1 + u^2)du = -1/u - u + u^3/3 + C
And that is the antiderivative of the given expression.