Indefinite Integrals – Basics & Constant of Integration

The integral of a constant k with respect to x is k*x + C. Here, the constant is 5. We can think of 5 as 5*x^0. Applying the power rule: ∫5*x⁰dx = 5 * (x^(0+1))/(0+1) + C = 5 * x^1 / 1 + C = 5x + C. We must include the constant of integration, C.

We know that the derivative of ln|x+1| is 1/(x+1), by the chain rule. For x+1 > 0, we can drop the absolute value. Therefore, ln(x+1) is an antiderivative of 1/(x+1). Option B differentiates to -2/(x+1)³. Option C differentiates to 2/(x+1)³. Option D's derivative requires the product rule and is not 1/(x+1).

When we find an antiderivative F(x) of a function f(x), we know that F'(x) = f(x). However, if we add any constant number C to F(x), the derivative remains f(x) because the derivative of a constant is zero. Therefore, the general antiderivative includes "+ C" to account for all possible vertical shifts of the function F(x). Without it, we would be missing many possible functions whose derivative is f(x).

We apply the power rule for integration, which states ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C for any real number n ≠ -1. We integrate each term separately. For the first term, 4x³: add 1 to the exponent (3+1=4) and divide by the new exponent: (4/4)x⁴ = x⁴. For the second term, 2x: this is 2x^1. Add 1 to the exponent (1+1=2) and divide by the new exponent: (2/2)x² = x². For the third term, -1: this is -1*x^0. Add 1 to the exponent (0+1=1) and divide by the new exponent: (-1/1)x^1 = -x. Finally, we add the constant of integration, C. The result is x⁴ + x² - x + C.

We integrate each term. The integral of eˣ is eˣ. The integral of the constant 7 is 7x. Therefore, the result is eˣ + 7x. We must add the constant of integration, C, so the final answer is eˣ + 7x + C.

The indefinite integral of a function f(x) is the collection of all antiderivatives of f(x). If F(x) is one antiderivative (so F'(x) = f(x)), then any other antiderivative can be written as F(x) + C, where C is any constant. Graphically, these are vertical shifts of the same curve. Therefore, they represent a family of functions differing by a constant.

First rewrite the integrand: 3√x = 3 * x^(½). Now apply the power rule: ∫3 * x^(½) dx = 3 * [x^( (½)+1 ) / ( (½)+1 )] + C = 3 * [x^(3/2) / (3/2)] + C. Dividing by (3/2) is the same as multiplying by (⅔). So, 3 * (⅔) * x^(3/2) + C = 2x^(3/2) + C.

We recall the derivative of cos(x) is -sin(x). Therefore, the derivative of -cos(x) is sin(x). This means an antiderivative of sin(x) is -cos(x). Including the constant of integration, the indefinite integral is -cos(x) + C.

We use the power rule in reverse for a composite function. We know that the derivative of (2x+5)⁴ is 4*(2x+5)³ * 2 = 8(2x+5)³. Notice our integrand is (2x+5)³. To get the antiderivative, we want to undo this. Since the derivative of (2x+5)⁴ gives 8(2x+5)³, we can write: ∫ (2x+5)³ dx = (1/8) * (2x+5)⁴ + C. We check by differentiating: d/dx [(1/8)(2x+5)⁴] = (1/8)*4*(2x+5)³*2 = (1/8)*8*(2x+5)³ = (2x+5)³.

Integrate term by term. For 10x⁴: add 1 to exponent (4+1=5), divide by 5: (10/5)x⁵ = 2x⁵. For -3 cos(x): the integral of cos(x) is sin(x), so ∫ -3 cos(x) dx = -3 sin(x). Combining and adding C gives 2x⁵ - 3 sin(x) + C.

By definition, finding an indefinite integral means finding a function whose derivative is the given function. This is the reverse operation of differentiation. Therefore, it is called anti-differentiation. Differentiation (A) is the process of finding a derivative. Optimization (C) involves finding maximum or minimum values. Simplification (D) is a general algebraic process.

We need to find a function whose derivative is x². The power rule for derivatives says the derivative of xⁿ is n*xⁿ⁻¹. To get x², we need an x³ term. Specifically, d/dx [(⅓)x³] = x². Adding a constant (like 5) does not change the derivative. Therefore, (⅓)x³ + 5 works. Option A differentiates to 9x². Option C differentiates to 2. Option D differentiates to 2x+2.

We recall the basic derivative rules. The derivative of sin(x) is cos(x). Therefore, by definition, an antiderivative of cos(x) is sin(x). We can verify this by differentiation: d/dx [sin(x)] = cos(x). Option A, -sin(x), differentiates to -cos(x). Option C, sec²(x), is the derivative of tan(x). Option D, -cos(x), differentiates to sin(x), not cos(x).

We integrate term by term. For the first term, x², we use the power rule: ∫x² dx = (⅓)x³ + C. For the second term, 1/x, we use the rule ∫(1/x) dx = ln|x| + C. Since the problem states x > 0, the absolute value is not needed, so ln(x) is correct. Combining these and including one constant of integration gives (⅓)x³ + ln(x) + C.

A closed-form antiderivative means we can write it using a finite combination of basic functions (polynomials, trig, exponential, logs, etc.). For e^(x²), mathematicians have proven that there is no such combination whose derivative is e^(x²). This is a known example of a function whose integral exists but cannot be expressed in elementary terms. It is continuous and differentiable, but its antiderivative is not an elementary function.