Integrals Ex 7.2: Mastering Integration by Substitution

Understanding the Method of Substitution

The method of substitution, often called u-substitution, is an essential technique for integrating composite functions or functions that involve a product of a function and its derivative. It's like a disguise for our integral: we transform a complex integral into a simpler one, which we can then solve using standard integration formulas. The core idea is to change the variable of integration from, say, 'x' to a new variable 't' (or 'u'). This change simplifies the integrand, making it recognizable as a basic integral.

Why is it needed? Imagine you have an integral like ∫ (2x + 5)^3 dx. You can't directly use ∫ x^n dx = (x^(n+1))/(n+1) + C because the base is not just 'x', but a linear function of 'x'. Similarly, for ∫ cos(2x) dx, the argument is '2x', not 'x'. The substitution method allows us to handle these cases systematically. It's particularly useful when an integral contains a function and its derivative (or a constant multiple of its derivative). For example, in ∫ f(g(x)) * g'(x) dx, if we let t = g(x), then dt = g'(x) dx, simplifying the integral to ∫ f(t) dt. This transformation makes the problem significantly easier to solve. It is one of the first advanced techniques you learn in integration and opens the door to solving a vast array of integral problems in physics, engineering, and other fields.

Step-by-Step Guide to Integration by Substitution

1. Identify the Substitution — Look for a part of the integrand (usually a function within another function, or the denominator of a fraction) whose derivative is also present (or a constant multiple of it) in the integral. Let this part be 't' (or 'u'). For example, if you have ∫ f(g(x)) * g'(x) dx, let t = g(x).
2. Differentiate and Find 'dt' — Differentiate your chosen 't' with respect to 'x' to find dt/dx. Then, express 'dx' in terms of 'dt' and 'x' (or usually, g'(x) dx becomes dt). For t = g(x), then dt/dx = g'(x), so dt = g'(x) dx.
3. Substitute into the Integral — Replace every 'x' term in the original integral with 't' terms, and 'dx' with its 'dt' equivalent. The entire integral should now be in terms of 't' only. If any 'x' terms remain, your substitution might be incorrect or incomplete.
4. Integrate with Respect to 't' — Solve the new integral using standard integration formulas. Remember to add the constant of integration, 'C', at this step.
5. Substitute Back 'x' — Finally, replace 't' with its original expression in terms of 'x' to get the answer in terms of the original variable. This is a common step that students sometimes forget!

Worked Examples of Substitution Method

• Example 1: Integrate ∫ 2x / (1 + x^2) dx Step 1: Identify the substitution. Notice that the derivative of (1 + x^2) is 2x. Let t = 1 + x^2. Step 2: Differentiate and find 'dt'. Differentiating t = 1 + x^2 with respect to x gives dt/dx = 2x. So, dt = 2x dx. Step 3: Substitute into the integral. The integral becomes ∫ 1/t dt. Step 4: Integrate with respect to 't'. ∫ 1/t dt = ln|t| + C. Step 5: Substitute back 'x'. Replace t with (1 + x^2). Final answer: ln|1 + x^2| + C. Example 2: Integrate ∫ sin(ax + b) dx Step 1: Identify the substitution. The argument of the sine function is (ax + b). Let t = ax + b. Step 2: Differentiate and find 'dt'. Differentiating t = ax + b with respect to x gives dt/dx = a. So, dx = dt/a. Step 3: Substitute into the integral. The integral becomes ∫ sin(t) (dt/a) = (1/a) ∫ sin(t) dt. Step 4: Integrate with respect to 't'. (1/a) ∫ sin(t) dt = (1/a) (-cos(t)) + C = - (1/a) cos(t) + C. Step 5: Substitute back 'x'. Replace t with (ax + b). Final answer: - (1/a) cos(ax + b) + C. Example 3: Integrate ∫ tan x dx Step 1: Rewrite tan x. We know tan x = sin x / cos x. So the integral is ∫ sin x / cos x dx. Step 2: Identify the substitution. The derivative of cos x is -sin x. Let t = cos x. Step 3: Differentiate and find 'dt'. Differentiating t = cos x with respect to x gives dt/dx = -sin x. So, dt = -sin x dx, or sin x dx = -dt. Step 4: Substitute into the integral. The integral becomes ∫ (1/t) * (-dt) = - ∫ (1/t) dt. Step 5: Integrate with respect to 't'. - ∫ (1/t) dt = -ln|t| + C. Step 6: Substitute back 'x'. Replace t with cos x. Final answer: -ln|cos x| + C. This can also be written as ln|sec x| + C.

Exam Tips and Common Mistakes in Substitution

Integration by substitution can be tricky if not approached carefully. Here are some critical tips to help you avoid common errors and score well:

• Choose the Right 't': This is often the most challenging step. A good heuristic is to choose 't' as the function whose derivative is also present (or can be easily adjusted with a constant) in the integrand. Often, it's the inner function of a composite function, or the denominator of a fraction. If your substitution makes the integral more complicated, or if you can't eliminate all 'x' terms, rethink your choice of 't'.
• Don't Forget 'dt': When you differentiate 't' with respect to 'x' to find dt/dx, make sure to write dt = (dt/dx) dx. Incorrectly transforming 'dx' is a very common mistake.
• Substitute ALL 'x' terms: After substitution, your integral must be entirely in terms of 't'. If any 'x' remains, your substitution is incomplete or incorrect.
• Substitute Back: Always remember to replace 't' with its original expression in terms of 'x' at the end. The final answer should be in terms of the original variable.
• Constant of Integration: Never forget to add the arbitrary constant 'C' after performing indefinite integration. This is an easy mark to lose.
• Practice Recognition: With enough practice, you'll start to recognize common patterns for substitution, making the process much faster. Work through all the examples in NCERT Ex 7.2 thoroughly.

Practice Questions with Solutions

• Q: Integrate ∫ (4x + 2)√(x^2 + x + 1) dx A: Step 1: Let t = x^2 + x + 1. Its derivative is dt/dx = 2x + 1. Step 2: So, dt = (2x + 1) dx. Notice that (4x + 2) = 2(2x + 1). Step 3: Substitute. The integral becomes ∫ 2√(t) dt = 2 ∫ t^(1/2) dt. Step 4: Integrate. 2 [t^(3/2) / (3/2)] + C = 2 (2/3) * t^(3/2) + C = (4/3) t^(3/2) + C. Step 5: Substitute back. (4/3) (x^2 + x + 1)^(3/2) + C. Final answer: (4/3) (x^2 + x + 1)^(3/2) + C
• Q: Integrate ∫ sin(tan⁻¹x) / (1 + x^2) dx A: Step 1: Let t = tan⁻¹x. Its derivative is dt/dx = 1/(1 + x^2). Step 2: So, dt = 1/(1 + x^2) dx. Step 3: Substitute. The integral becomes ∫ sin(t) dt. Step 4: Integrate. -cos(t) + C. Step 5: Substitute back. -cos(tan⁻¹x) + C. Final answer: -cos(tan⁻¹x) + C
• Q: Integrate ∫ e^(tan⁻¹x) / (1 + x^2) dx A: Step 1: Let t = tan⁻¹x. Its derivative is dt/dx = 1/(1 + x^2). Step 2: So, dt = 1/(1 + x^2) dx. Step 3: Substitute. The integral becomes ∫ e^t dt. Step 4: Integrate. e^t + C. Step 5: Substitute back. e^(tan⁻¹x) + C. Final answer: e^(tan⁻¹x) + C
• Q: Integrate ∫ (sec^2(7 - 4x)) dx A: Step 1: Let t = 7 - 4x. Its derivative is dt/dx = -4. Step 2: So, dx = dt/(-4). Step 3: Substitute. The integral becomes ∫ sec^2(t) (dt/-4) = (-1/4) ∫ sec^2(t) dt. Step 4: Integrate. (-1/4) tan(t) + C. Step 5: Substitute back. (-1/4) tan(7 - 4x) + C. Final answer: (-1/4) tan(7 - 4x) + C

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