NCERT Class 12 Maths Chapter 7 Exercise 7.8: Definite Integrals

Theoretical Foundation: The Second Fundamental Theorem of Calculus

To solve problems in integrals ex 7 8 class 12 ncert, we rely heavily on the Second Fundamental Theorem of Calculus. This theorem provides a direct link between differentiation and integration. If $f(x)$ is a continuous function defined on the closed interval $[a, b]$, and $F(x)$ is any anti-derivative (indefinite integral) of $f(x)$ such that $F'(x) = f(x)$, then the definite integral is given by:

$\int_{a}^{b} f(x)\,dx = [F(x)]_{a}^{b} = F(b) - F(a)$

Here, '$a

$(F(b) + C) - (F(a) + C) = F(b) - F(a)$

Thus, the constant $C$ always cancels out, making the final value unique and constant.

Method of Evaluation: Substitution and Limits

1. Find the Indefinite Integral First — Temporarily ignore the limits of integration. Integrate the integrand $f(x)$ using standard formulas, substitution, partial fractions, or integration by parts to find the anti-derivative $F(x)$.
2. Perform Substitution and Change Limits (If Applicable) — If you substitute a new variable $t = g(x)$, you MUST change the limits of integration. Calculate the new lower limit $t_1 = g(a)$ and upper limit $t_2 = g(b)$. Evaluate the integral with respect to $t$ using these new limits, eliminating the need to substitute back to $x$ at the end.
3. Apply the Limits of Integration — Evaluate the anti-derivative at the upper limit and subtract its value at the lower limit: $F(b) - F(a)$.

Step-by-Step Worked Examples

• Example 1: Evaluate $\int_{1}^{2} (3x^2 + 2x + 1)\,dx$ Step 1: Find the general anti-derivative of the integrand. $F(x) = \int (3x^2 + 2x + 1)\,dx = x^3 + x^2 + x$ Step 2: Apply the Second Fundamental Theorem of Calculus with limits from 1 to 2. $[F(x)]_{1}^{2} = (2^3 + 2^2 + 2) - (1^3 + 1^2 + 1)$ Step 3: Simplify calculations. $= (8 + 4 + 2) - (1 + 1 + 1) = 14 - 3 = 11$. Final Answer: 11
• Example 2: Evaluate $\int_{0}^{\pi/4} \tan^2 x\,dx$ Step 1: Express $\tan^2 x$ in terms of a standard integrable function using the identity $\tan^2 x = \sec^2 x - 1$. Step 2: Set up the limits on the modified expression: $\int_{0}^{\pi/4} (\sec^2 x - 1)\,dx = [\tan x - x]_{0}^{\pi/4}$ Step 3: Substitute limits: $= (\tan(\pi/4) - \pi/4) - (\tan(0) - 0) = (1 - \pi/4) - (0) = 1 - \pi/4$. Final Answer: $1 - \frac{\pi}{4}$

Avoid Common Mistakes in Definite Integrals

Many CBSE students lose marks due to simple errors in Exercise 7.8 topics. Keep these rules in mind:

1. Always Change Limits During Substitution: If you substitute $t = u(x)$, calculate the new limits. Integrating with original limits after substitution is a recipe for errors.
2. Be Careful with Signs: The formula is $F(b) - F(a)$. If $F(a)$ has negative terms, distribute the negative sign carefully (e.g., $F(b) - (-F(a)) = F(b) + F(a)$).
3. Domain Continuity: Make sure the function $f(x)$ is continuous in the interval $[a, b]$. The Fundamental Theorem cannot be applied directly if there is a point of discontinuity inside the interval.

Practice Questions with Solutions

• Q: Evaluate $\int_{2}^{3} \frac{1}{x}\,dx$ A: Step 1: Find the anti-derivative of $f(x) = \frac{1}{x}$. We know that $\int \frac{1}{x}\,dx = \log|x|$. Step 2: Apply the limits of integration from 2 to 3. $[\log|x|]_{2}^{3} = \log(3) - \log(2)$. Step 3: Simplify using logarithm properties: $\log(a) - \log(b) = \log(\frac{a}{b})$. Final answer: $\log\left(\frac{3}{2}\right)$
• Q: Evaluate $\int_{0}^{\pi/2} \cos x\,dx$ A: Step 1: Find the anti-derivative of $\cos x$. The integral is $\sin x$. Step 2: Apply the upper and lower limits. $[\sin x]_{0}^{\pi/2} = \sin(\pi/2) - \sin(0)$. Step 3: Substitute the trigonometric values: $\sin(\pi/2) = 1$ and $\sin(0) = 0$. $= 1 - 0 = 1$. Final answer: 1
• Q: Evaluate $\int_{0}^{1} \frac{1}{1 + x^2}\,dx$ A: Step 1: Identify the standard integration formula. We know that $\int \frac{1}{1+x^2}\,dx = \tan^{-1}x$. Step 2: Apply the limits of integration from 0 to 1. $[\tan^{-1}x]_{0}^{1} = \tan^{-1}(1) - \tan^{-1}(0)$. Step 3: Substitute the inverse trigonometric values. $= \frac{\pi}{4} - 0 = \frac{\pi}{4}$. Final answer: $\frac{\pi}{4}$
• Q: Evaluate $\int_{0}^{1} x e^{x^2}\,dx$ using substitution. A: Step 1: Let $t = x^2$. Differentiating gives $dt = 2x\,dx$, which means $x\,dx = \frac{dt}{2}$. Step 2: Change limits. When $x = 0$, $t = 0^2 = 0$. When $x = 1$, $t = 1^2 = 1$. Step 3: Rewrite and integrate in terms of $t$: $\int_{0}^{1} e^t \frac{dt}{2} = \frac{1}{2} [e^t]_{0}^{1} = \frac{1}{2} (e^1 - e^0) = \frac{1}{2} (e - 1)$. Final answer: $\frac{e - 1}{2}$

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