Calculus Integrals Quiz

The integral of tdt from a to b is equal to -1 times the integral of tdt from b to a. This relationship highlights how changing integration limits impacts the integral's value.

The integral of dx / (9 - 4x^2) from 0 to 1/3 is equal to (1/6) * arcsin(2x/3) + C, where C is the constant of integration.

The antiderivative of tan(x) - 1 with respect to x is log(sec(x)) + C. This involves finding a function whose derivative is the given expression.

If the derivative of f(x) with respect to x is 2x + x^3 and f(1) = 1, then f(x) = x^2 + 3ln|x|. This is achieved by integrating the derivative and using the initial condition.

The integral (#xe^x - 1 + e^(x - 1) ) / (xe^x + e^x) dx from P = 1/4 to 1 is equal to (1/e) * ln(xe^x + e^x) + C. This is obtained by using substitution to simplify the integrand.

The integral of dx / (x(x^7 + 1)) from 0 to 1 is equal to (1/7) * ln(x^7 + 1) + C, where C is the constant of integration. Solve this by substituting x^7 = t and simplifying.

The integral of cos(x^2) dx from 0 to pi/2 is equal to (1/2) * sqrt(pi)/2. This is a special integral that is solved using the Gaussian integral.

The value of the definite integral (#cot(q)*cosec^2(q)dq) from p/4 to p/2 is equal to 1/2. Solve by substituting cot(q) = t and simplifying the integral.

The value of the definite integral (#cos(x^2)dx) from 0 to pi/2 is equal to 1/2 * sqrt(pi)/2. This type of integral requires special methods to solve and involves a key relationship between cos(x^2) and Gaussian integral.

The integral of (x * cos(x) + sin(x) + 1) dx from -2 to 2 is equal to 4. Utilize integration by parts, and simplify the integrated terms.

The definite integral of f(x) dx from -a to a is 0 if f(x) is an odd function. This rule applies when the integrand exhibits symmetry about the origin.

The sum of the definite integral of f(x) dx from a to b and from b to a is 0. This reflects the cancellation of areas when integrating in opposite directions.

The definite integral of x dx from 0 to 1 is equal to 1/2. This is a simple integral that is solved using the power rule of integration.

The integral of (1/(1 + x^2)) * dx from 0 to 1 is equal to pi/4. This integral involves the inverse tangent function and can be solved by substituting x = tan(t).

The definite integral of |1 - x| dx from 0 to 2 is equal to 1. This integral involves the absolute value function and requires breaking it into separate cases.

The integral of log(2x) dx is equal to x * log(2x) - x + C, where C is the constant of integration. This is obtained by using integration by parts, taking u = log(2x) and dv = dx.

The integral of cot(q)*cosec(q)dq is equal to -cosec(q) + C, where C is the constant of integration. This is a standard integral derived using the derivative of cosec(q).

The integral of sin(x)*x dx is equal to -cos(x)*x + sin(x) + C. This is solved using integration by parts, where u = x and dv = sin(x)dx.

The integral of cosec(x)*cot(x)dx is equal to -cosec(x) + C, where C is the constant of integration. This can be solved directly by recognizing the derivative of cosec(x).

The definite integral of log(cos(x)) dx from 0 to pi/2 needs a special approach. This involves applying the property of definite integrals, #f(x)dx = #f(a + b - x)dx from a to b, to transform the integral and solve it.

The definite integral of sin^2(x) / cos^8(x) dx from 0 to pi/4 is equal to 1/7 * (tan^7(x) + C), where C is the constant of integration. This is obtained using substitution of tan(x) as t and simplifying the integral.

The integral of sin(x + cos^2(x)) * (sin(x) + 2sin(x)cos(x)) dx is equal to x + C, where C is the constant of integration. This is solved using the substitution u = sin(x) + cos^2(x) and simplifying the integral.

The integral of sec^2(x) dx is equal to tan(x) + C. This is a standard integral that can be directly identified by recalling the derivative of tan(x).

The integral of dx / (x^2 + 16) is equal to (1/4) * arctan(x/4) + C, where C is the constant of integration. This can be solved by using the arctangent function or by completing the square in the denominator and applying a suitable substitution.

The definite integral of (2 - 3sin(x)) /cos^2(x) dx from 0 to pi/2 is equal to 1. This is solved by separating terms and using trigonometric identities to simplify the integral.

The integral of sin^n(x) dx from 0 to pi/2 is a special case of the Wallis formula. The formula provides a way to evaluate this integral based on the value of ‘n’.

The integral of dx / (1 + x^n)^(1/n) from 0 to 1 is equal to the integral of tan^(2/n-1)(q) * dq from 0 to p/4. This is solved by using a specific substitution and simplifying the integral.

The definite integral of sin^3(x) dx from 0 to 2*pi is equal to 0, using the property of definite integrals that states if f(x) is odd, then #f(x) dx = 0 from -a to a.

The definite integral of sin(3x)*cos(5x) dx from 0 to pi/2 is equal to (1/16) * (cos(2x) - cos(8x)) + C, where C is the constant of integration. This can be solved using the product-to-sum trigonometric identity and then integrating.

The definite integral of f(x) dx from -a to a is equal to 2 * (definite integral of f(x) dx from 0 to a) if f(x) is an even function. This applies to symmetrical functions.

The definite integral of (sin(x) + cos(x)) dx from -pi/2 to pi/2 is equal to 2. This can be solved using the property of even functions in definite integrals.

The definite integral of x^5 dx from a to b is equal to (1/6) * (b^6 - a^6). Apply the power rule of integration and evaluate the limits.

The integral of (tan^-1(x))^2 / (1 + x^2) dx from 0 to 1 is equal to (pi^2 / 192). This is solved using the substitution tan^-1(x) = t and simplifying the integral.

The integral of sin^2(x) * cos^2(x) dx from 0 to pi/4 is equal to (pi/32). This is solved using the double angle formula for cos(2x) and simplifying the integral.

The definite integral of (1 - x) dx from 0 to 1 is equal to 1/2, while the definite integral of (x - 1) dx from 1 to 2 is also equal to 1/2. This is solved by breaking the integral into two parts due to the absolute value and then integrating each part separately.