BCA Semester 2 – Mathematics II – Unit 4: Integration and Its Applications (8Hrs)

Question 1

Evaluate: \[ \quad \text{a) } \int0^2 \frac{dx}{x^2-25} \] \[ \quad \text{b) } \int \frac{2x+5}{x^2+5x} \, dx \]

Marks: 5

Year: 2022 Final TU FOHSS

$\textbf{(a) Evaluation of } \displaystyle \int0^2 \frac{dx}{x^2 - 25}$ First, factor the denominator: \[ x^2 - 25 = (x - 5)(x + 5) \] Using partial fractions: \[ \frac{1}{(x - 5)(x + 5)} = \frac{1}{1

Question 2

Evaluate the integrals: \[ \text{a) } \int \frac{2x+5}{\sqrt{x^2+5x}}\,dx \] \[ \text{b) } \int e^{2x}\log x \, dx \]

Marks: 5

Year: 2023 Final TU FOHSS

(a) Evaluation of $ \displaystyle \int \frac{2x+5}{\sqrt{x^2+5x}}\,dx $ Notice that the numerator is the derivative of the expression under the square root: \[ \frac{d}{dx} (x^2 + 5x) = 2x + 5 \] He

Question 3

Evaluate the following integrals: \[ \quad \text{a) } \int \frac{2x+3}{x^2+3x}\,dx \] \[ \quad \text{b) } \int e^x \cos x \, dx \]

Marks: 5

Year: 2024 Final TU FOHSS

(a) Integration using partial fractions \[ \int \frac{2x+3}{x^2+3x}\,dx = \int \frac{2x+3}{x(x+3)}\,dx \] Decompose into partial fractions: \[ \frac{2x+3}{x(x+3)} = \frac{A}{x} + \frac{B}{x+3} \] Mult