Integrals (Indefinite and Definite) Solved Examples (12 Mathematics)
Integrals find antiderivatives and areas under curves. These examples cover power rule, trigonometric integrals, definite integrals, and applications like
TL;DR: Integrals find antiderivatives and areas under curves. These examples cover power rule, trigonometric integrals, definite integrals, and applications…
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Integrals find antiderivatives and areas under curves. These examples cover power rule, trigonometric integrals, definite integrals, and applications like
Integrals (Indefinite and Definite) — Solved Numerical Examples (Step by Step)
Example 1: Find the integral of 5x^4 with respect to x.
Solution: Use power rule: integral of x^n = x^(n+1) / (n+1) + C
integral of 5x^4 = 5 × integral of x^4
= 5 × [x^5 / 5] + C
= x^5 + C
Example 2: Find integral of sin(x) from 0 to pi.
Solution: integral from 0 to pi of sin(x) dx
Antiderivative of sin(x) = -cos(x)
= [-cos(x)] from 0 to pi
= -cos(pi) - (-cos(0))
= -(-1) - (-1)
= 1 + 1 = 2
Example 3: Evaluate integral of (3x^2 + 2x + 1) dx.
Solution: Use power rule on each term:
integral of 3x^2 dx = 3 × x^3/3 = x^3
integral of 2x dx = 2 × x^2/2 = x^2
integral of 1 dx = x
Total = x^3 + x^2 + x + C
Example 4: Find the area under the curve y = x^2 from x = 0 to x = 3.
Solution: Area = integral from 0 to 3 of x^2 dx
Antiderivative of x^2 = x^3/3
Area = [x^3/3] from 0 to 3
= (3^3/3) - (0^3/3)
= 27/3 - 0
= 9 square units
Example 5: Evaluate integral of e^x from 0 to 1.
Solution: Antiderivative of e^x = e^x
integral from 0 to 1 of e^x dx = [e^x] from 0 to 1
= e^1 - e^0
= e - 1
≈ 2.718 - 1 = 1.718
Example 6: Find integral of cos(x) with respect to x.
Solution: Antiderivative of cos(x) = sin(x) + C
integral of cos(x) dx = sin(x) + C
Example 7: Evaluate integral of 1/x from 1 to e.
Solution: Antiderivative of 1/x = ln|x|
integral from 1 to e of (1/x) dx = [ln|x|] from 1 to e
= ln(e) - ln(1)
= 1 - 0 = 1
Tips
- Power rule: integral of x^n = x^(n+1)/(n+1) + C (for n ≠ -1).
- Special cases: integral of 1/x = ln|x|, integral of e^x = e^x.
- For definite integrals, evaluate antiderivative at upper limit minus lower limit.
- When integrating sum/difference, integrate each term separately.
Frequently Asked Questions
What is the constant of integration C?
The constant C represents all antiderivatives of a function. Since derivative of any constant is 0, multiple functions (differing by a constant) have the same derivative. C is added to indefinite integrals; not needed in definite integrals.
What is the difference between definite and indefinite integrals?
Indefinite integral finds the antiderivative function plus a constant C. Definite integral calculates the area under the curve between two specific limits and gives a numerical value.
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