Indefinite Integral

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What is the result of integrating 2sin(2θ) with respect to θ?

−2(1 − cos(2θ)) + C

−2θ + sin(2θ) + C (correct)

−2tan(θ) + C

−2θ + cos(2θ) + C

What is the formula for integration by parts?

∫udv = u∫vdx + ∫v∫udx

∫udv = ∫uvdx - ∫vdx∫udx

∫udv = uv - ∫vdu

∫udv = u∫vdx - ∫dv∫udx (correct)

What is the purpose of choosing the first and second functions in integration by parts?

To ensure the correct result (correct)

To make the calculation easier

To simplify the integral

To make a proper choice of substitution

What is the result of integrating sin(x) with respect to x?

−cos(x) + C Signup and view all the answers

What is the purpose of rationalization of the denominator in indefinite integrals?

To simplify the integral Signup and view all the answers

What is the result of integrating 1−x² with respect to x?

x + sin⁻¹(x) + C Signup and view all the answers

What is the primary purpose of dividing the numerator and denominator by x^2 in the given integral?

To simplify the integrand Signup and view all the answers

What is the result of substituting x^2 + 2 = t^2 - 2 into the original integral?

∫(1/(t^2 + 2)) dt Signup and view all the answers

Which of the following integrals is equivalent to the original integral?

∫(x^4 + kx^2 + 1)/(x^2 + 2x + 1) dx Signup and view all the answers

What is the purpose of the substitution x^2 + 2 = t^2 - 2?

To simplify the integrand and evaluate the integral Signup and view all the answers

What is the general technique used to evaluate the given integral?

Trigonometric substitution Signup and view all the answers

What is the result of integrating a function f(x) with respect to x?

A function φ(x) such that d(φ(x)) = f(x)dx Signup and view all the answers

What happens to the degree of a polynomial function when it is differentiated?

It decreases by 1 Signup and view all the answers

What is the result of differentiating an integral?

The integrand itself Signup and view all the answers

What is the correct trigonometric identity for sin mx cos nx?

1/2 [sin(m + n)x + sin(m - n)x] Signup and view all the answers

If I = ∫sin x(1 - cos x) dx, what is the correct substitution to evaluate this integral?

Substitute cos x = t Signup and view all the answers

What is the correct reduction formula for the integral ∫sin mx.dx?

-(1/m)cos mx + C Signup and view all the answers

If I = ∫cos mx.sin nx dx, what is the correct trigonometric identity to use?

1/2 [sin(m + n)x + sin(m - n)x] Signup and view all the answers

If I = ∫cos mx.cos nx dx, what is the correct trigonometric identity to use?

1/2 [cos(m + n)x + cos(m - n)x] Signup and view all the answers

What is the form of the integral after substituting $x - \alpha = t$?

$\int a + r cos(t) dt$ Signup and view all the answers

What is the integral of sin x with respect to x?

-cos x + C Signup and view all the answers

What is the form of the given integral after simplifying?

$\int a + r cos(x - \alpha) dx$ Signup and view all the answers

What is the integral of cos x with respect to x?

sin x + C Signup and view all the answers

What is the integral of sec x tan x with respect to x?

sec x + C Signup and view all the answers

What is the integral of cosec x cot x with respect to x?

-cosec x + C Signup and view all the answers

What is the integral of tan x with respect to x?

log |sec x| + C Signup and view all the answers

What is the integral of cot x with respect to x?

log |sin x| + C Signup and view all the answers

What is the integral of sec^2 x with respect to x?

tan x + C Signup and view all the answers

What is the integral of cosec^2 x with respect to x?

-cot x + C Signup and view all the answers

What is the integral of sin^2 x with respect to x?

x/2 - sin(2x)/2 + C Signup and view all the answers