Integration

Step 1: Use product-to-sum identities.
Step 2: Integrate term by term.

Step 1: Division. (x²)/(x²-1)=1+(1)/(x²-1)
Step 2: Integrate. dx+(1)/(x²-1)dx

Step 1: Use substitution for integration.
We use appropriate trigonometric substitution to solve the integral.
Step 2: Conclusion.
The integral evaluates to . Final Answer:

Step 1: Write denominator using :
Step 2: Put , then simplify.
Step 3: Final integration gives

Integrating by parts,

Thus .

Differentiating -(eˣ)/(1+sin x) gives (eˣcos x)/(1+sin x). Hence the integral equals C-(eˣ)/(1+sin x).

Given conditions imply f(x)=x(x-1)². Thus, (f(x))/(x³-1)=1 ⟹ int (f(x))/(x³-1)dx=x+C

N/A

By inspection, differentiating (eˣ)/(1+sin x) yields a numerator involving cos x-1 after simplification. Hence, int (cos x-1)/(sin x+eˣ)dx = C-(eˣ)/(1+sin x)

The integral is a standard form. Recognize that integrates to . Here, , so: Alternatively, use substitution: let , then , and the integral becomes: Thus, option (2) is correct.