Integrals: NCERT Exercise 7.1 Concepts and Solutions

Understanding Indefinite Integrals (Antiderivatives)

In differentiation, you are given a function F(x) and asked to find its derivative f(x). For example, if F(x) = x³, its derivative is f(x) = 3x². Integration, or finding the antiderivative, reverses this. You are given the derivative f(x) = 3x² and asked to find the original function F(x). But there's a small catch. The derivative of x³ is 3x², but so is the derivative of x³ + 5, x³ - 100, or any function x³ + C (where C is a constant). Because the derivative of any constant is zero, we can't know the exact original constant. This is why we always add a 'Constant of Integration', + C, to our answer. The notation for an indefinite integral is:

∫f(x) dx = F(x) + C

Here:

• ∫ is the integral symbol.
• f(x) is the integrand (the function to be integrated).
• dx indicates that the integration is with respect to the variable x.
• F(x) is the antiderivative of f(x).
• C is the constant of integration.

Fundamental Integration Formulas for Exercise 7.1

Power Rule

∫xⁿ dx = (xⁿ⁺¹ / (n+1)) + C, where n ≠ -1

Constant Rule

∫k dx = kx + C, where k is a constant.

Reciprocal Rule

∫(1/x) dx = log|x| + C

Exponential Rule

∫eˣ dx = eˣ + C and ∫aˣ dx = (aˣ / log a) + C

Trigonometric Formulas

∫sin(x) dx = -cos(x) + C ∫cos(x) dx = sin(x) + C ∫sec²(x) dx = tan(x) + C ∫csc²(x) dx = -cot(x) + C ∫sec(x)tan(x) dx = sec(x) + C ∫csc(x)cot(x) dx = -csc(x) + C

Linearity of Integration

∫[k f(x) ± g(x)] dx = k ∫f(x) dx ± ∫g(x) dx

Worked Examples for Integrals Ex 7.1

• Question: Find the integral of ∫(4x³ - 6) dx Step 1: Apply Linearity. Break the integral into two separate integrals: ∫(4x³ - 6) dx = ∫4x³ dx - ∫6 dx Step 2: Use the Constant Multiple Rule. Move the constants outside the integral sign: = 4 ∫x³ dx - 6 ∫1 dx Step 3: Apply the Power Rule and Constant Rule. For the first term, use ∫xⁿ dx = xⁿ⁺¹/(n+1). Here, n=3. For the second term, use ∫k dx = kx. = 4 (x³⁺¹ / (3+1)) - 6x = 4 (x⁴ / 4) - 6x Step 4: Simplify and add the constant of integration. = x⁴ - 6x + C Final Answer: The integral of (4x³ - 6) is x⁴ - 6x + C.
• Question: Find the integral of ∫(2x² - 3sin(x) + 5√x) dx Step 1: Apply Linearity. Separate the integral into three parts: ∫2x² dx - ∫3sin(x) dx + ∫5√x dx Step 2: Rewrite the terms and move constants out. Remember that √x = x¹/². = 2∫x² dx - 3∫sin(x) dx + 5∫x¹/² dx Step 3: Integrate each term using the appropriate formula. - For x², use the power rule (n=2). - For sin(x), ∫sin(x) dx = -cos(x). - For x¹/², use the power rule (n=1/2). = 2(x³/3) - 3(-cos(x)) + 5(x¹/²⁺¹ / (1/2+1)) Step 4: Simplify and add the constant of integration. = (2/3)x³ + 3cos(x) + 5(x³/² / (3/2)) = (2/3)x³ + 3cos(x) + 5 * (2/3)x³/² = (2/3)x³ + 3cos(x) + (10/3)x³/² + C Final Answer: (2/3)x³ + 3cos(x) + (10/3)x³/² + C.
• Question: Find the integral of ∫(x³ + 3x + 4) / √x dx Step 1: Simplify the integrand. Before integrating, divide each term in the numerator by the denominator, √x = x¹/². (x³ + 3x + 4) / x¹/² = x³/x¹/² + 3x¹/x¹/² + 4/x¹/² Using the law of exponents xᵃ/xᵇ = xᵃ⁻ᵇ: = x³⁻¹/² + 3x¹⁻¹/² + 4x⁻¹/² = x⁵/² + 3x¹/² + 4x⁻¹/² Step 2: Integrate the simplified expression. Now integrate term by term: ∫(x⁵/² + 3x¹/² + 4x⁻¹/²) dx = ∫x⁵/² dx + 3∫x¹/² dx + 4∫x⁻¹/² dx Step 3: Apply the Power Rule to each term. = [x⁵/²⁺¹ / (5/2+1)] + 3[x¹/²⁺¹ / (1/2+1)] + 4[x⁻¹/²⁺¹ / (-1/2+1)] = [x⁷/² / (7/2)] + 3[x³/² / (3/2)] + 4[x¹/² / (1/2)] Step 4: Simplify and add the constant of integration. = (2/7)x⁷/² + 3(2/3)x³/² + 4(2)x¹/² + C = (2/7)x⁷/² + 2x³/² + 8x¹/² + C Final Answer: (2/7)x⁷/² + 2x³/² + 8√x + C.

Exam Tip: Never Forget '+C'!

One of the most common mistakes students make in indefinite integration is forgetting to add the constant of integration, + C, at the end of their answer. Remember, the integral ∫f(x) dx represents a family of functions whose derivative is f(x), not just a single function. The + C accounts for all the possible vertical shifts of the antiderivative curve. In board exams, you can lose a crucial half or full mark for this omission, even if the rest of your integration is perfect. Make it a habit: as soon as you finish integrating, write + C.

Practice Questions with Solutions

• Q: Find the antiderivative of f(x) = (x - 1/x)². A: Step 1: First, expand the expression. f(x) = x² - 2(x)(1/x) + (1/x)² = x² - 2 + 1/x² = x² - 2 + x⁻². Step 2: Now, integrate the simplified expression term by term: ∫(x² - 2 + x⁻²) dx = ∫x² dx - ∫2 dx + ∫x⁻² dx. Step 3: Apply the power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) and the constant rule. (x³/3) - 2x + (x⁻¹/(-1)) + C. Step 4: Simplify the result. (x³/3) - 2x - 1/x + C. Final answer: The antiderivative is x³/3 - 2x - 1/x + C.
• Q: Integrate ∫(ax² + bx + c) dx with respect to x. A: Step 1: Use the linearity property of integrals to separate the terms: ∫ax² dx + ∫bx dx + ∫c dx. Step 2: Move the constants (a, b, c) outside the integral signs: a∫x² dx + b∫x¹ dx + c∫1 dx. Step 3: Apply the power rule to x² and x¹, and the constant rule to 1: a(x³/3) + b(x²/2) + c(x). Step 4: Combine the terms and add the constant of integration, C. (a/3)x³ + (b/2)x² + cx + C. Final answer: (a/3)x³ + (b/2)x² + cx + C.
• Q: Find ∫(sec²(x) - 1) dx. A: Step 1: Recognize that sec²(x) - 1 = tan²(x). However, integrating tan²(x) directly is not a basic formula. It's easier to integrate sec²(x) and 1 separately. Step 2: Use the linearity property: ∫(sec²(x) - 1) dx = ∫sec²(x) dx - ∫1 dx. Step 3: Apply the standard integration formulas: ∫sec²(x) dx = tan(x) and ∫1 dx = x. Step 4: Combine the results and add the constant of integration: tan(x) - x + C. Final answer: tan(x) - x + C.
• Q: If the derivative of a function is f'(x) = 4x³ - 3/x⁴ and f(2) = 0, find the function f(x). A: Step 1: To find f(x), we need to integrate f'(x). So, f(x) = ∫(4x³ - 3x⁻⁴) dx. Step 2: Integrate term by term: f(x) = 4∫x³ dx - 3∫x⁻⁴ dx = 4(x⁴/4) - 3(x⁻³/(-3)) + C. Step 3: Simplify the expression: f(x) = x⁴ + x⁻³ + C = x⁴ + 1/x³ + C. Step 4: Use the given condition f(2) = 0 to find C. f(2) = 2⁴ + 1/2³ + C = 16 + 1/8 + C = 0. This gives 129/8 + C = 0, so C = -129/8. Final answer: f(x) = x⁴ + 1/x³ - 129/8.

Frequently Asked Questions