Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 2 Integral Calculus I Ex 2.9 Text Book Back Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Ex 2.9

Evaluate the following using properties of definite integrals:

Question 1.

\(\int_{-π/4}^{π/4}\) x³ cos³ x dx

Solution:

Let f(x) = x³cos³x

f(-x) = (-x)³ cos³(-x)

= -x³ cos³x

f(-x) = -f(x)

⇒ f(x) is an odd function

∴ \(\int_{-π/4}^{π/4}\) x³ cos³ x dx = 0

Question 2.

\(\int_{-π/2}^{π/2}\) sin² θ dθ

Solution:

Let f(θ)= sin² θ

f(-θ) = sin² (-θ) = [sin (-θ)]²

= [-sin θ]² = sin² θ

f(-θ) = f(θ)

∴ f(θ) is an even function

Question 3.

\(\int_{-1}^{1}\) log(\(\frac { 2-x }{2+x}\)) dx

Solution:

Question 4.

\(\int_{0}^{π/2}\) \(\frac { sin^7x }{sin^7x+cos^7x}\) dx

Solution:

Using the property

\(\int_{0}^{a}\) f(x) dx = \(\int_{0}^{a}\) f(a – x) dx

Question 5.

\(\int_{0}^{1}\) log (\(\frac { 1 }{x}\) – 1) dx

Solution:

Question 6.

\(\int_{0}^{1}\) \(\frac { x }{(1-x)^{3/4}}\) dx

Solution:

Let I = \(\int_{0}^{1}\) log \(\frac { x }{(1-x)^{3/4}}\) dx

Using the property

\(\int_{0}^{a}\) f(x) dx = \(\int_{0}^{a}\) f(a – x) dx

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