Q. 52 - Calculus

Question

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that f and g are continuous functions and that k is any real number.

$\int \left(f^{\text{'}}\left(x\right)g\left(x\right)+f\left(x\right)g^{\text{'}}\left(x\right)\right)dx=...$

Step-by-Step Solution

Answer

The complete integration rule is $\int \left(f^{\text{'}}\left(x\right)g\left(x\right)+f\left(x\right)g^{\text{'}}\left(x\right)\right)dx=f\left(x\right)g\left(x\right)+\mathrm{C}$. 

Step 1. Given information.

Consider the given indefinite integral is $\int \left(f^{\text{'}}\left(x\right)g\left(x\right)+f\left(x\right)g^{\text{'}}\left(x\right)\right)dx=...$.

Step 2. Fill the blank for the indefinite integral ∫ f ' x g x + f x g ' x d x .

The theorem of Reversing the Product rule for Indefinite Integrals states that, $\int \left(f^{\text{'}}\left(x\right)g\left(x\right)+f\left(x\right)g^{\text{'}}\left(x\right)\right)dx=f\left(x\right)g\left(x\right)+\mathrm{C}$.

Therefore, it can be said the appropriate fill in black will be $\int \left(f^{\text{'}}\left(x\right)g\left(x\right)+f\left(x\right)g^{\text{'}}\left(x\right)\right)dx=f\left(x\right)g\left(x\right)+\mathrm{C}$.