Determine the following integrals: (a) ∫x^4 d x (f) ∫5^x d x (b) ∫3 sinx d x (g) ∫x^-4 d x (c

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Determine the following integrals: (a) ∫x^4 d x (f) ∫5^x d...

Determine the following integrals: (a) $\int x^{4} \mathrm{~d} x$ (f) $\int 5^{x} \mathrm{~d} x$ (b) $\int 3 \sin x d x$ (g) $\int x^{-4} \mathrm{~d} x$ (c) $\int 4 . e^{x} \mathrm{~d} x$ (h) $\int \frac{4}{x} \mathrm{~d} x$ (d) $\int 6 \mathrm{~d} x$ (i) $\int 3 \cos x \mathrm{~d} x$ (e) $\int 3 x^{\frac{1}{2}} \mathrm{~d} x$ (j) $\int 2 \sec ^{2} x d x$

Determine the following integrals:
(a) $\int x^{4} \mathrm{~d} x$
(f) $\int 5^{x} \mathrm{~d} x$
(b) $\int 3 \sin x d x$
(g) $\int x^{-4} \mathrm{~d} x$
(c) $\int 4 . e^{x} \mathrm{~d} x$
(h) $\int \frac{4}{x} \mathrm{~d} x$
(d) $\int 6 \mathrm{~d} x$
(i) $\int 3 \cos x \mathrm{~d} x$
(e) $\int 3 x^{\frac{1}{2}} \mathrm{~d} x$
(j) $\int 2 \sec ^{2} x d x$

Key Concepts

Constant Multiple Rule

The constant multiple rule in integration states that a constant factor can be pulled out of the integral. That is, the integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function. This property simplifies the integration process, especially when dealing with functions that are scaled by a constant coefficient.

Integration of Trigonometric Functions

Trigonometric integrals involve functions such as sin x, cos x, and sec^2 x, and are essential in various areas of calculus. The antiderivatives for these functions are well-known: the integral of sin x yields -cos x; the integral of cos x yields sin x; and the integral of sec^2 x yields tan x, each plus an arbitrary constant. Mastery of these antiderivatives is crucial for solving problems in both pure and applied mathematics.

Integration Leading to Logarithmic Functions

When integrating functions that yield a logarithm, a classic example is the integral of 1/x, where the antiderivative is ln|x| plus a constant. This concept is closely related to the power rule but is treated as a special case since the power rule does not apply when the exponent is -1. Understanding this concept is important for managing integrals involving rational functions.

Power Rule for Integration

The power rule is a fundamental technique used for integrating polynomial functions. It states that for any real number n ? -1, the integral of x^n with respect to x is x^(n+1)/(n+1) plus a constant of integration. This rule simplifies the process of finding antiderivatives for functions expressed as a sum of terms with powers of x.

Integration of Exponential Functions

Exponential functions have unique properties that allow for straightforward integration. For example, the integral of the natural exponential function e^x is e^x plus a constant. For exponential functions with any positive base a (where a ? 1), the integral of a^x involves dividing by the natural logarithm of the base, resulting in a^x/ln(a) plus the constant of integration. This technique is widely applicable in many problems involving growth and decay.

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