AP ECET MATHEMATICS Multiple Choice Questions (MCQs)

106. What is the indefinite integral of \( \sin(x) \)?

A) \( -\cos(x) + C \)

B) \( \cos(x) + C \)

C) \( \sin(x) + C \)

D) \( -\sin(x) + C \)

Answer: A) \( -\cos(x) + C \)

107. The integral of \( x^2 \) with respect to \( x \) is:

A) \( \frac{x^3}{3} + C \)

B) \( \frac{x^2}{2} + C \)

C) \( x^3 + C \)

D) \( x^2 + C \)

Answer: A) \( \frac{x^3}{3} + C \)

108. Which of the following is the standard form for the indefinite integral of \( e^x \)?

A) \( e^x + C \)

B) \( e^x - C \)

C) \( \ln(x) + C \)

D) \( \frac{e^x}{x} + C \)

Answer: A) \( e^x + C \)

109. The integral of \( \frac{1}{x} \) with respect to \( x \) is:

A) \( \ln(x) + C \)

B) \( \ln|x| + C \)

C) \( \frac{1}{x} + C \)

D) \( x \ln(x) + C \)

Answer: B) \( \ln|x| + C \)

110. Integration of \( \sin^2(x) \) using the reduction formula results in:

A) \( \frac{x}{2} - \frac{\sin(2x)}{4} + C \)

B) \( \frac{x}{2} + \frac{\sin(2x)}{4} + C \)

C) \( -\frac{\cos(2x)}{2} + C \)

D) \( -\frac{x}{2} + \frac{\cos(2x)}{4} + C \)

Answer: A) \( \frac{x}{2} - \frac{\sin(2x)}{4} + C \)

111. The integral of \( \cos(x) \) is:

A) \( \sin(x) + C \)

B) \( -\sin(x) + C \)

C) \( \cos(x) + C \)

D) \( -\cos(x) + C \)

Answer: A) \( \sin(x) + C \)

112. Which method of integration is used to solve \( \int x e^x dx \)?

A) Integration by substitution

B) Integration by parts

C) Integration by trigonometric identities

D) Direct integration

Answer: B) Integration by parts

113. What is the integral of \( \frac{1}{x^2 + 1} \)?

A) \( \tan^{-1}(x) + C \)

B) \( \ln(x) + C \)

C) \( \frac{1}{2} \ln(x^2 + 1) + C \)

D) \( \frac{1}{x} + C \)

Answer: A) \( \tan^{-1}(x) + C \)

114. The integral of \( e^{x^2} \) with respect to \( x \) is:

A) \( \text{No elementary function exists} \)

B) \( e^{x^2} + C \)

C) \( x e^{x^2} + C \)

D) \( \frac{e^{x^2}}{2x} + C \)

Answer: A) \( \text{No elementary function exists} \)

115. What is the integral of \( \frac{1}{\sqrt{x^2 - 1}} \)?

A) \( \ln|x + \sqrt{x^2 - 1}| + C \)

B) \( \sin^{-1}(x) + C \)

C) \( \tan^{-1}(x) + C \)

D) \( \cos^{-1}(x) + C \)

Answer: A) \( \ln|x + \sqrt{x^2 - 1}| + C \)

116. The integral of \( \frac{1}{\sqrt{1 - x^2}} \) is:

A) \( \ln(1 - x^2) + C \)

B) \( \sin^{-1}(x) + C \)

C) \( \cos^{-1}(x) + C \)

D) \( \tan^{-1}(x) + C \)

Answer: B) \( \sin^{-1}(x) + C \)

117. Which technique is used to evaluate the integral \( \int \frac{1}{x^2 + a^2} dx \)?

A) Integration by substitution

B) Trigonometric substitution

C) Direct integration

D) Integration by parts

Answer: B) Trigonometric substitution

118. The integral of \( \ln(x) \) is:

A) \( x \ln(x) - x + C \)

B) \( x \ln(x) + C \)

C) \( x + \ln(x) + C \)

D) \( x^2 \ln(x) + C \)

Answer: A) \( x \ln(x) - x + C \)

119. The integral of \( \frac{x}{x^2 + 1} \) is:

A) \( \ln(x^2 + 1) + C \)

B) \( \tan^{-1}(x) + C \)

C) \( \frac{x^2}{2} + C \)

D) \( \frac{1}{x} + C \)

Answer: B) \( \tan^{-1}(x) + C \)

120. In integration by parts, which of the following is the correct formula?

A) \( \int u \, dv = uv - \int v \, du \)

B) \( \int u \, dv = uv + \int v \, du \)

C) \( \int u \, dv = u + \int v \, du \)

D) \( \int u \, dv = u - \int v \, du \)

Answer: A) \( \int u \, dv = uv - \int v \, du \)