Grade 12 Mathematics Quiz: Indefinite Integrals with Logarithmic and Trigonometric Functions

Find the indefinite integral of sin(x) with respect to x.

• x^2 + C

• 1/cos(x) + C

• -cos(x) + C

• sin(x) + C

Answer explanation

The indefinite integral of sin(x) with respect to x is -cos(x) + C. This is the correct choice because the derivative of -cos(x) + C is sin(x).

Evaluate the integral of cos(2x) with respect to x.

• (1/2)sin(2x) + C

• 2sin(x) + C

• sin(2x) + C

• (1/2)cos(2x) + C

Answer explanation

The integral of cos(2x) with respect to x is (1/2)sin(2x) + C. This is the correct choice because the derivative of (1/2)sin(2x) + C is cos(2x).

Calculate the integral of tan(x) with respect to x.

• e^x + C

• cos(x) + C

• ln|sec(x)| + C

• sin(x) + C

Answer explanation

The integral of tan(x) with respect to x is ln|sec(x)| + C. This is the correct choice because the derivative of ln|sec(x)| is tan(x).

Determine the indefinite integral of sec^2(x) with respect to x.

• sin(x) + C

• csc(x) + C

• tan(x) + C

• cos(x) + C

Answer explanation

The indefinite integral of sec^2(x) is tan(x) + C. This is because the derivative of tan(x) is sec^2(x), and the integral is the reverse of the derivative.

Find the integral of ln(x) with respect to x.

• ln(x) + C

• x^2*ln(x) + C

• x*ln(x) - x + C

• x*ln(x) + C

Answer explanation

The integral of ln(x) with respect to x is x*ln(x) - x + C. This choice correctly represents the antiderivative of ln(x).

Evaluate the indefinite integral of 1/x with respect to x.

• ln|x| + C

• 2ln|x| + C

• x^2 + C

• 1/2x^2 + C

Answer explanation

The indefinite integral of 1/x with respect to x is ln|x| + C. This choice correctly represents the antiderivative of the given function.

Calculate the integral of ln(2x) with respect to x.

• x*ln(2x) - x + C

• x*ln(2x) + C

• 2x*ln(x) - x + C

• ln(2x) - x + C

Answer explanation

The integral of ln(2x) with respect to x is x*ln(2x) - x + C. This choice correctly represents the antiderivative of the given function.

Determine the integral of arctan(x) with respect to x.

• x*arctan(x) + C

• arctan(x) + C

• x*arctan(x) - 0.5*ln(1+x^2) + C

• x^2*arctan(x) - ln(1+x^2) + C

Answer explanation

The integral of arctan(x) with respect to x is x*arctan(x) - 0.5*ln(1+x^2) + C. This choice is correct because it matches the given integral.

Find the indefinite integral of arcsin(x) with respect to x.

• x*arcsin(x) + sqrt(1-x^2) + C

• arcsin(x) + C

• x*arcsin(x) - sqrt(1-x^2) + C

• x^2*arcsin(x) + C

Answer explanation

The indefinite integral of arcsin(x) with respect to x is x*arcsin(x) + sqrt(1-x^2) + C. This choice correctly accounts for the integral of arcsin(x) and the square root term.

Evaluate the integral of arccos(x) with respect to x.

• x*arccos(x) + sqrt(1-x^2) + C

• x^2*arccos(x) + C

• arccos(x) + C

• x*arccos(x) - sqrt(1-x^2) + C

Answer explanation

The integral of arccos(x) with respect to x is x*arccos(x) + sqrt(1-x^2) + C. This choice is correct because it matches the given integral.