RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 7 Integrals Ex 7.8

Question 1.
\(\int_{a}^{b} \)x dx
Answer:

Question 2.
\(\int_{0}^{5}\) (x + 1) dx
Answer:

Question 3.
\(\int_{2}^{3}\) x² dx
Answer:

Question 4.
\(\int_{1}^{4}\) (x² - x) dx
Answer:
Let \(\int_{a}^{b}\) f(x) dx = \(\int_{1}^{4}\) (x² - x) dx
Here, f(x) = x² - x, a = 1, b = 4,
(b - a) = (4 - 1)3,
∴ nh = 3
f(a) = f(1) = 1² - 10
f(a + h) =f(1 + h) = (1 + h)² - (1 + h)
= 1 + 2h + h² - 1 - h = h² + h
f(a + 2h) = f(1 + 2h)
= (1 + 2h)² - (1 + 2h)
= 1 + 2.2h + 4h² - 1 - 2h
= 4h² + 2h
f(a + 3h) = f(1 + 3h)
= (1 + 3h)² - (1 + 3h)
= 1 + 2.3h + 9h² - 1 - 3h
= 9h² + 3h
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f[a + (n - 1)h] = f[1 + (n - 1)h]
= [1 + (n - 1)h]² - [1 + (n - 1)h]
= 1 + 2(n - 1)h + (n - 1)² h² - 1 - (n - 1)h
= (n - 1)² h² + (n - 1)h

Question 5.
\(\int_{-1}^{1}\) e^x dx
Answer:
Let \(\int_{a}^{b}\) f(x) dx = \(\int_{-1}^{1}\) e^x dx
Here, f(x) = e^x, a= - 1, b = 1
and nh = b - a = {1 - (- 1)} = 1 + 1 = 2
f(a) = f(- 1) = e^-1
f(a + h) = f(- 1 + h) = e^{-1 + h}
f(a + 2h) = f(- 1 + 2h) = e^{-1 + 2h}
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f{a + (n - 1)h} = f{- 1 + (n - 1)h}
= e^{- 1 + (n - 1)h}

Question 6.
\(\int_{0}^{4}\) (x + e^2x) dx
Answer:
Let \(\int_{a}^{b}\) f(x) dx = \(\int_{0}^{4}\) (x + e^2x) dx
Here, f(x) = x + e^2x
a = 0, b = 4, nh = (4 - 0) = 4
∵ f(a) = f(0) = 0 + e⁰ = 1
f(a + h) = f(0 + h) = h + e^2h
f(a + 2h) = f(0 + h) = h + e^4h
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f{a + (n - 1)h} = f{0 + (n - 1)h}
= (n - 1)h + e^{2(n - 1)h}