Indefinite Integrals – Advanced Techniques & Applications

The notation Integral of f(x) dx = F(x) + C implies that F(x) is the antiderivative of f(x). By the fundamental definition of antiderivatives, this means that if you take the derivative of the result F(x), you must obtain the integrand f(x).

We apply the Power Rule for Integration. The exponent is -3. We add 1 to the exponent to get -2. Then, we divide the coefficient 7 by this new exponent -2. This yields -(7/2)x^(-2) + C.

We need to recognize standard derivative forms. We know that the derivative of sec(x) is sec(x)tan(x). Since the constant 5 is just a multiplier, the antiderivative of 5 sec(x)tan(x) is simply 5 times the antiderivative of sec(x)tan(x), which results in 5 sec(x) + C.

The rule for integrating an exponential function with a base other than e (bˣ) is bˣ / ln(b) + C. This is because the derivative of bˣ is bˣ * ln(b). To reverse the differentiation, we must divide by the natural log of the base. Here the base is 2, so the integral is 2ˣ / ln(2) + C.

This is a power rule application with a linear inner function. First, apply the power rule to the outer function: add 1 to the exponent (3+1=4) and divide by the new exponent (4). This gives (1/4)(4x - 1)⁴. However, we must also divide by the coefficient of x in the linear term (which is 4) to reverse the chain rule. So we have (1/4) * (1/4) * (4x - 1)⁴, which simplifies to (1/16)(4x - 1)⁴ + C.

Before integrating, simplify the expression by dividing each term in the numerator by the denominator. (x² + 1)/x becomes x + 1/x. Now integrate each term individually. The integral of x is x²/2. The integral of 1/x is ln|x|. Combining these gives x²/2 + ln|x| + C.

When we differentiate a composite function like sin(5x), we use the Chain Rule to get cos(5x) * 5. When we integrate cos(5x), we are reversing this process. We divide by 5 to "undo" the multiplication that happened during differentiation. Therefore, the logic for handling linear composites in integration is derived from reversing the Chain Rule.

It is best to expand the square first. (√(x) + 2)² expands to x + 4√(x) + 4. Rewriting the radical as an exponent gives x + 4x^(½) + 4. Now integrate term by term. The integral of x is x²/2. For 4x^(½), add 1 to the exponent (3/2) and divide by 3/2 (multiply by 2/3), resulting in 4*(⅔)x^(3/2) = (8/3)x^(3/2). The integral of 4 is 4x. Combining these yields x²/2 + (8/3)x^(3/2) + 4x + C.

We can rewrite the integrand using trigonometric identities. sin(x)/cos²(x) can be separated into (1/cos(x)) * (sin(x)/cos(x)). This simplifies to sec(x)tan(x). We recognize sec(x)tan(x) as the derivative of sec(x). Therefore, the integral is sec(x) + C.

We are integrating an exponential function with a linear exponent (-x, where the coefficient is -1). The integral of e^(ax) is (1/a)e^(ax). Here, a = -1. So we take the constant 3, multiply by the integral of e⁻ˣ which is -e⁻ˣ, resulting in -3e⁻ˣ + C.

This is a logarithmic integral form. The integral of 1/u is ln|u|. Here, u = 4 - x. Because the variable x has a negative sign (coefficient is -1), we must divide by -1 to reverse the chain rule. Thus, the result is -1 * ln|4 - x| + C, or -ln|4 - x| + C.

We apply the Power Rule to each term. For x⁴, add 1 to the exponent and divide by 5 to get x⁵/5. For -3x², add 1 to the exponent (3) and divide by 3: -3x³/3 simplifies to -x³. For the constant 5, the integral is 5x. Combining these gives x⁵/5 - x³ + 5x + C.

Solving the differential equation for y means finding the indefinite integral of the right side with respect to x. We integrate 6x⁵ to get 6(x⁶/6), which simplifies to x⁶. We integrate eˣ to get eˣ. Adding the constant of integration, the general solution is y = x⁶ + eˣ + C.

Rewrite the integrand with a negative exponent: (3x + 1)^(-3). Apply the power rule: add 1 to the exponent to get -2, and divide by the new exponent -2. This gives -(½)(3x + 1)^(-2). However, because of the linear inner function 3x+1, we must also divide by the slope 3. So we have -(½) * (⅓) * (3x + 1)^(-2). This simplifies to -1 / [6(3x + 1)²] + C.

This is a standard integral that corresponds to an inverse trigonometric function. We recall from differentiation rules that the derivative of arctan(x) (or inverse tangent) is exactly 1 / (1 + x²). Therefore, the antiderivative is arctan(x) + C.