Q.Evaluate the definite integral: $\int_{0}^{\pi/4} (2 \sec^2 x + x^3 + 2) \, dx$

The key idea is to split the integral into simpler parts using linearity, then apply standard antiderivatives.

Step 1: Split the integral

Step 2: Integrate each term

• $\int \sec^2 x \, dx = \tan x$, so $2\int_{0}^{\pi/4} \sec^2 x \, dx = 2[\tan x]_{0}^{\pi/4} = 2(1 - 0) = 2$ …

We are asked to evaluate

The key idea is that the integral of a sum is the sum of the integrals. Each term here has a straightforward antiderivative — no symmetry tricks needed, just direct integration. Let’s go term by term.

1. Integrate $2\sec^2 x$ Recall that $\frac{d}{dx}(\tan x) = \sec^2 x$. So

2. Integrate $x^3$ Using the power rule $\int x^n dx = \frac{x^{n+1}}{n+1}$ for $n \neq -1$:

3. Integrate the constant $2$

Now combine these antiderivatives. An antiderivative of the whole integrand is

4. Evaluate from $0$ to $\pi/4$ By the Fundamental Theorem of Calculus:

Compute $F(\pi/4)$:

Hence

Compute $F(0)$: …

Method: Use linearity to integrate a sum term by term

The integral of a sum is the sum of the integrals, and constant factors pull outside. This turns any polynomial-plus-standard-function integrand into a set of elementary antiderivatives.

Steps

Step 1: Break the integrand along $+$ and $-$ signs.

Step 2: Apply the standard antiderivative for each term. …

Common Mistakes

Mistake 1: Misremembering $\int\sec^2x\,dx$.

Why it's wrong: $\int\sec^2x\,dx=\tan x$, not $\sec x$ or $\tan^2 x$. Correct approach: recall $\frac{d}{dx}\tan x=\sec^2x$ and evaluate $2\tan x$ at the limits.

Mistake 2: Mishandling the power $(\pi/4)^4$. …