Q.48 ∫₀¹ z dx + ∫₁² (2 − x) dx = ________.

The expression in the question is mathematically incomplete as shown in the image, so the exact integral cannot be reconstructed with certainty. The symbol that looks like
∫₀¹ 2x dx + ∫ (2 − x) dx is not fully visible and the limits (and even the variable in the second integral) are cut off. Because of this, no unique numerical value or “correct option” can be determined from the snippet provided.

Reconstructing what is visible

From the scanned image, the following parts can be read:

• A definite integral sign with lower limit 0 and upper limit 1.
• An integrand that appears as 2x or possibly 2z.
• A second integral added to the first, whose integrand looks like (2 − x).
• The right‑hand side is a blank line for the final answer.

A common style of such questions in calculus exams is:

∫₀¹ 2x dx + ∫₁² (2 − x) dx = ?

If that is the intended question, then:

∫₀¹ 2x dx = [x²]₀¹ = 1

∫₁² (2 − x) dx = [2x − x²/2]₁² = (4 − 2) − (2 − 1/2) = 2 − 3/2 = 1/2

So the sum would be 1 + 1/2 = 3/2. However, this is only valid if the second integral really has limits 1 to 2. Those limits are not visible in the image, so this reconstruction is speculative.

Why the options cannot be explained

You asked for a detailed explanation of every option, but the image does not show any options (A, B, C, D) or the complete integral.

Without:

• Clear integrands (whether it is 2x, 2z, or something similar).
• The exact upper and lower limits of both integrals.
• The multiple‑choice options.

No rigorous solution or option‑wise explanation is possible, because small changes in limits or integrand (for example 2x vs x², or limits 0 to 1 vs 0 to 2) completely change the answer.

How to solve such double‑integral questions in general

When you see an expression of the type ∫_a^b f(x) dx + ∫_b^c g(x) dx, use these steps:

1. Check limits carefully.
   Ensure the first integral runs from a to b and the second from b to c. If both integrals are of the same function f(x), you can combine them as
   ∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx.
2. Integrate each part separately.
   Find antiderivatives F(x) of each integrand and use the Fundamental Theorem of Calculus: ∫_a^b f(x) dx = F(b) − F(a).
3. Add numerical results.
   Compute each definite integral, then sum the results to get the final value.
4. Use geometry when possible.
   For linear functions like 2x or 2 − x, interpret the integrals as areas of triangles or trapeziums under straight lines, which often makes calculations faster and helps to cross‑check the answer.

Because the integral in the image is incomplete, any SEO article with a precise slug, keyphrase, and meta description referring to a specific numeric answer would risk being mathematically wrong.

Once you provide a clear text version of the question (for example: “Evaluate ∫₀¹ 2x dx + ∫₁² (2 − x) dx”), an SEO‑friendly structure could look like this:

Introduction example:
In this article, learn how to evaluate ∫₀¹ 2x dx + ∫₁² (2 − x) dx using basic integration rules and a simple geometric approach that is perfect for competitive exams.