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Chapter 7: Problem 6

How would you evaluate \(\int \cos ^{2} x \sin ^{3} x d x ?\)

Short Answer

Expert verified
Question: Evaluate the integral $\int \cos^2 x \sin^3 x dx$. Answer: $-\left(\frac{\cos^3 x}{3} - \frac{\cos^5 x}{5}\right) + C$

Step by step solution

01

Identify the substitution

To make the integration simpler, we can apply a substitution method. We can recognize that the derivative of \(\cos x\) is \(- \sin x\) and the derivative of \(\sin x\) is \(\cos x\). Therefore, it would be convenient to choose the substitution \(u = \cos x\) so that \(-du = \sin x dx\).
02

Applying substitution and simplification

We substitute \(u = \cos x\) and find the new expression for the integral. Using this substitution, the integral becomes: $$ \int \cos^2 x \sin^3 x dx = \int u^2 (-\sin^2 x) \cdot \sin x \,dx = -\int u^2 (1 - \cos^2 x) du $$ Now, substitute \(\cos x = u\) and \(\sin^2 x = 1 - u^2\): $$ -\int u^2 (1 - u^2) du = -\int (u^2 - u^4) du $$
03

Integrate the new expression

Now, we can integrate the simplified expression to find the solution: $$ -\int (u^2 - u^4) du = -\left(\frac{u^3}{3} - \frac{u^5}{5}\right) + C $$ Here, C is the constant of integration.
04

Re-substitute the original variable

Finally, we substitute back the original variable \(x\) by putting \(u = \cos x\): $$ -\left(\frac{u^3}{3} - \frac{u^5}{5}\right) + C = -\left(\frac{\cos^3 x}{3} - \frac{\cos^5 x}{5}\right) + C $$ So the final answer is: $$ \int \cos^2 x \sin^3 x dx = - \left( \frac{\cos^3 x}{3} - \frac{\cos^5 x}{5} \right) + C $$

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Most popular questions from this chapter

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Use the indicated methods to solve the following problems with nonuniform grids. A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, its vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The vertical velocities at selected times are shown in the table (with units of \(\mathrm{ft} / \mathrm{min}\) ). $$\begin{array}{|l|c|c|c|c|c|c|c|} \hline t \text { (min) } & 0 & 1 & 1.5 & 3 & 3.5 & 4 & 5 \\ \hline \begin{array}{l} \text { Velocity } \\ \text { (ft/min) } \end{array} & 0 & 100 & 120 & 150 & 110 & 90 & 80 \\ \hline \end{array}$$ a. Use the Trapezoid Rule to estimate the elevation of the balloon after five minutes. Remember that the balloon starts at an elevation of \(5400 \mathrm{ft}\) b. Use a right Riemann sum to estimate the elevation of the balloon after five minutes. c. A polynomial that fits the data reasonably well is $$g(t)=3.49 t^{3}-43.21 t^{2}+142.43 t-1.75$$ Estimate the elevation of the balloon after five minutes using this polynomial.

Evaluating an integral without the Fundamental Theorem of Calculus Evaluate \(\int_{0}^{\pi / 4} \ln (1+\tan x) d x\) using the following steps. a. If \(f\) is integrable on \([0, b],\) use substitution to show that $$ \int_{0}^{b} f(x) d x=\int_{0}^{b / 2}(f(x)+f(b-x)) d x $$ b. Use part (a) and the identity \(\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}\) to evaluate \(\int_{0}^{\pi / 4} \ln (1+\tan x) d x\) (Source: The College Mathematics Journal 33, 4, Sep 2004).
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