Evaluate the following indefinite integrals.
i.du/u²
ii. 6dv/v⅝
In both case apply the Power Rule:
∫ xⁿ dx = ( x ⁿ ⁺ ¹ ) / ( n +1 ) + C
∫ du / u²
∫ du / u² = ∫ u ⁻² ∙ du= ( u ⁻² ⁺ ¹ ) / ( - 2 +1 ) + C =
u ⁻¹ / ( - 1 ) + C = - 1 / u + C
∫ 6 dv / v⅝ = 6 ∫ dv / v⅝ = 6 ∫ v ⁻ ⅝ dv = 6 v ⁻ ⅝ ⁺ ¹ / ( - 5 / 8 + 1 ) + C =
6 v ⅜ / ( 3 / 8 ) + C = 6 ∙ 8 ∙ v ⅜ / 3 + C = 3 ∙ 2 ∙ 8 ∙ v ⅜ / 3 + C =
2 ∙ 8 ∙ v ⅜ + C = 16 ∙ v ⅜ + C
To evaluate these indefinite integrals, we will use the power rule for integration.
i. ∫ du/u²:
The power rule states that ∫ x^n dx = (x^(n+1))/(n+1), where n is any real number except -1.
In this case, we have ∫ du/u², where n = -2. Adding 1 to -2 gives -1, so we should end up with -1 in the denominator. Thus, applying the power rule, we can rewrite the integral as ∫ u^(-2) du.
Now, applying the power rule, we get ∫ u^(-2) du = (u^(-2 + 1))/(-2 + 1) = -u^(-1) = -1/u + C, where C is the constant of integration.
Therefore, the indefinite integral of du/u² is -1/u + C.
ii. ∫ 6dv/v^(5/8):
Again, we will use the power rule for integration.
In this case, we have ∫ 6dv/v^(5/8), where n = 5/8. Applying the power rule, we get ∫ v^(5/8) dv = (v^(5/8 + 1))/(5/8 + 1) = (v^(13/8))/(13/8) = 8/13 * v^(13/8) + C, where C is the constant of integration.
Therefore, the indefinite integral of 6dv/v^(5/8) is 8/13 * v^(13/8) + C.