Question

Evaluate the following integrals: (i) \(\int \frac{3 \sin x+2 \cos x}{3 \cos x+2 \sin x} d x\) (ii) \(\int \frac{\cos x d x}{2 \sin x+3 \cos x}\) (iii) \(\int \frac{4 \sin x+5 \cos x}{5 \sin x+4 \cos x} d x\) (iv) \(\int \frac{6+3 \sin x+14 \cos x}{3+4 \sin x+5 \cos x} d x\)

Short answer

Based on the step-by-step solution provided, the short answer for each integral is as follows: (i) \(\int \frac{3 \sin x+2 \cos x}{3 \cos x+2 \sin x} d x = \ln |3 \cos x + 2 \sin x|+C\) (ii) \(\int \frac{\cos x d x}{2 \sin x+3 \cos x} = \ln |2\sin x+3\cos x|+C\) (iii) \(\int \frac{4 \sin x+5 \cos x}{5 \sin x+4 \cos x} d x = \ln |5\sin x+4\cos x|+C\) (iv) \(\int \frac{6+3 \sin x+14 \cos x}{3+4 \sin x+5 \cos x} d x = \ln |(3+4\sin x+5\cos x)^8|+C\)

Step-by-step solution

Make a substitution

Let \(u = 3 \cos x + 2 \sin x\). Then, \(-\sin x u' = -2 \cos x + du\), which simplifies to \(du= 3 \sin x + 2 \cos x + du'\sin x\). Now, the integral now becomes: \(\int \frac{du}{u}\)

Integrate

The integral now becomes \(\ln |u|+C\).

Substitute back

Substitute back \(u=3 \cos x + 2 \sin x\), so the final answer is \(\ln |3 \cos x + 2 \sin x|+C\). #(ii) Integral 2#

Make a substitution

Let \(u=2\sin x+3\cos x\). Then, \(du=(3\sin x-2\cos x)dx\). Now, the integral becomes: \(\int \frac{du}{u}\)

Integrate

The integral now becomes \(\ln |u|+C\).

Substitute back

Substitute back, \(u=2\sin x+3\cos x\), so the final answer is \(\ln |2\sin x+3\cos x|+C\). #(iii) Integral 3#

Make a substitution

Let \(u=5\sin x+4\cos x\). Then, \(du=(-4\sin x+5\cos x)dx\). Now, the integral becomes: \(\int \frac{du}{u}\)

Integrate

The integral now becomes \(\ln |u|+C\).

Substitute back

Substitute back, \(u=5\sin x+4\cos x\), so the final answer is \(\ln |5\sin x+4\cos x|+C\). #(iv) Integral 4#

Make a substitution

Let \(u=3+4\sin x+5\cos x\). Then, \(du=(-5\sin x+4\cos x)dx\). Now, the integral of \(\frac{6+3\sin x+14\cos x}{u}\) turns into: \(\int \frac{2du+du'+6du'-u'}{u}\)

Integrate

The integral now becomes \(2\ln |u|+(6\int \frac{du}{u})\). Integrate to get: \(2\ln |u|+6\ln |3+4\sin x+5\cos x|+C\)

Simplify

Use the logarithm properties to simplify the final answer: \(\ln |(3+4\sin x+5\cos x)^8|+C\)

Key concepts

Integration Techniques

Understanding different integration techniques is crucial when solving calculus problems. Integral calculus is used to determine quantities like area, volume, and other quantities when the opposite process of differentiation is applied. There are numerous techniques of integration that can be employed, including the power rule, integration by parts, trigonometric substitution, and partial fractions, among others.

One common approach is the substitution method, also known as u-substitution, which is particularly useful when dealing with composite functions or integrals of products of functions. By substituting variables and simplifying the integrand, the problem can often be transformed into a simpler integral that is easier to evaluate. Once the integral is solved in terms of the new variable, a back-substitution is performed to express the solution in terms of the original variable.

Another valuable technique involves the integration of trigonometric functions, which often involves applying trigonometric identities to simplify the integral before solving. Recognizing patterns and the right method to use can make seemingly complex integrals much more straightforward to calculate.

Substitution Method

The substitution method, or u-substitution, is a powerful tool for simplifying and solving integrals. The goal of the substitution technique is to transform a complicated integral into a simpler one by changing the variable of integration.

The process typically involves three key steps: First, identify a part of the integrand that can be substituted with a new variable, which we'll denote as u. Then, differentiate u with respect to x to find du and rewrite the integral in terms of u and du. Integrate with respect to u, and effectively, the integrand becomes much easier to work with. Finally, once the integral in terms of u is found, the original variable is substituted back to express the answer in the original terms. It's essential to choose the right substitution to simplify the integral effectively, and sometimes, this might require some trial and error.

Applying this method often simplifies the integral into a standard form, such as \(\int\frac{du}{u}\), which has the readily recognizable solution \(\ln |u|+C\).

Integral of Trigonometric Functions

Integrals that involve trigonometric functions require special attention due to the periodic and oscillatory nature of sin, cos, tan, and other trigonometric functions. To integrate these functions effectively, we often rely on trigonometric identities, which help in simplifying the integrands.

For example, we might use the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\) to convert an integral involving \(\sin(x)\) and \(\cos(x)\) into a more manageable form. Similarly, double-angle, half-angle, and angle-sum identities frequently turn complicated-looking integrals into basic ones that can be solved readily with standard integration techniques.

In the exercise provided, integrals of functions that are ratios of trigonometric expressions are addressed through substitution. By recognizing that the numerator can be written as a derivative of the denominator, we apply the substitution method to simplify each integral to \(\int\frac{du}{u}\), which is a standard form and straightforward to integrate, leading to the natural logarithmic function as a solution.