Question

How do you find the antiderivative of ${(1 + \cos x)^2}$?

Answer 1

Start by substituting ${\cos ^2}x = \dfrac{1}{2}(1 + \cos 2x)$. Substitute the values in place of the terms to make the equation easier to solve. Then we will differentiate the term. Now we will substitute these terms in the original expression and integrate.

Complete step by step solution:
First we will start off by substituting ${\cos ^2}x = \dfrac{1}{2}(1 + \cos 2x)$.
Now we differentiate this term to form a proper equation and for substituting the
terms.
Now we will first open the brackets by squaring the given expression. Also, remember that ${(a + b)^2} = {a^2} + 2ab + {b^2}$.
$= \int {{{(1 + \cos x)}^2}dx} \\\ = \int {(1 + {{\cos }^2}x + 2\cos x)dx} \\\$
Now we will distribute the integral within the terms.

$$= \int {(1 + {{\cos }^2}x + 2\cos x)dx} \\\ = \int {1dx} + \int {{{\cos }^2}xdx} + \int {2\cos xdx} \\\$$

Now if there are any integers then take them out of the interval.

$$= \int {1dx} + \int {{{\cos }^2}xdx} + \int {2\cos xdx} \\\ = \int {1dx} + \int {{{\cos }^2}xdx} + 2\int {\cos xdx} \\\$$

Now we will integrate each of the terms separately.
So, first we evaluate the values of the integrals.
$\int {1dx} = x + a$
$\int {{{\cos }^2}xdx} = \dfrac{x}{2} + \sin x\cos x + b$
$2\int {\cos xdx} = 2\sin x + c$
Hence, the antiderivative of ${(1 + \cos x)^2}$ will be $x + 2\sin x + \dfrac{x}{2} + \sin x\cos x + d$
Additional Information:
A derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations. In general, scientists observe changing systems to obtain rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions.

Note: While substituting the terms make sure you are taking into account the degrees and signs of the terms as well. While applying the power rule make sure you have considered the power with their respective signs. Remember the identity ${\cos ^2}x = \dfrac{1}{2}(1 + \cos 2x)$.