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Chapter 17: Problem 8

Find \(\int \frac{\sin ^{3} x}{\cos x}+\sin x \cos x \mathrm{~d} x\).

Short Answer

Expert verified
Question: Integrate the following function: \(\int \frac{\sin^3 x}{\cos x}+\sin x \cos x \mathrm{~d} x\) Answer: The integral of the given function is: \(-\cos x + \frac{1}{2}\sin^2 x + C\)

Step by step solution

01

Divide the integrand into two parts

The integrand can be divided into two parts as follows: \[ \frac{\sin^3x}{\cos x} + \sin x \cos x \]
02

Integrate \(\frac{\sin^3x}{\cos x}\) with substitution

Let's use the substitution method to integrate \(\frac{\sin^3x}{\cos x}\). Let \(u = \sin x\), so \(du = \cos x dx\). Then, the first integral can be written as: \[ \int \frac{u^3}{\sqrt{1-u^2}} du \] Next, use a second substitution to solve the integral. Let \(v = 1 - u^2\), so \(dv = -2u du\). We have: \[ -\frac{1}{2} \int \frac{dv}{\sqrt{v}} \] Now, integrate with respect to \(v\): \[ -\frac{1}{2} \cdot 2\sqrt{v} + C_1 = -\sqrt{1 - u^2} + C_1 \] Replace \(u = \sin x\) to find the first part of our solution: \[ -\sqrt{1 - \sin^2 x} + C_1 = -\sqrt{\cos^2 x} + C_1 = -\cos x + C_1 \]
03

Integrate \(\sin x \cos x\)

Now, we need to find the integral of the second part, which is given as \(\sin x \cos x\). Using the substitution method again, let \(w = \sin x\), so \(dw = \cos x dx\). Rewrite the integral as: \[ \int w \, dw \] Then, integrate with respect to \(w\): \[ \frac{1}{2}w^2 + C_2 = \frac{1}{2}\sin^2 x + C_2 \]
04

Add both integrals

Finally, we add the two integrals we found in steps 2 and 3: \[ -\cos x + C_1 + \frac{1}{2}\sin^2 x + C_2 = -\cos x + \frac{1}{2}\sin^2 x + C \] Therefore, the requested integral is: \[ \int \frac{\sin^3 x}{\cos x}+\sin x \cos x \mathrm{~d} x = -\cos x + \frac{1}{2}\sin^2 x + C \]

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Most popular questions from this chapter

Find \(\int \frac{x}{\sqrt{4-x^{2}}} \mathrm{~d} x\).

Find the area enclosed by the graph of \(y=\frac{1}{\sqrt{9-4 t^{2}}}\) between \(t=0\) and \(t=1\).

By writing \(\ln x\) as \(1 \times \ln x\) find \(\int \ln x \mathrm{~d} x\).

Find (a) \(\int x \sin (2 x) \mathrm{d} x\), (b) \(\int t \mathrm{e}^{3 t} \mathrm{~d} t\) (c) \(\int x \cos x \mathrm{~d} x\).

Find the following integrals using an appropriate technique selected from this chapter: (a) \(\int \frac{1}{4+x^{2}} \mathrm{~d} x\) (b) \(\int \cos ^{2} x \mathrm{~d} x\) (c) \(\int \frac{1}{4-t} \mathrm{~d} t\) (d) \(\int_{0}^{\pi} \cos t \mathrm{~d} t\) (e) \(\int_{1}^{1.2} \tan x \mathrm{~d} x\) (f) \(\int \frac{2 x^{2}+x+2}{x^{3}+x} \mathrm{~d} x\) (g) \(\int \sin 3 t \cos 3 t \mathrm{~d} t\) (h) \(\int \frac{1}{16-x^{2}} \mathrm{~d} x\) (i) \(\int \frac{3}{\sqrt{t^{2}+0.25}} \mathrm{~d} t\) (j) \(\int_{-1}^{1} \cosh 3 t \mathrm{~d} t\) (k) \(\int_{1}^{3} \frac{1}{1+x^{2}} \mathrm{~d} x\) (1) \(\int \frac{1}{\sqrt{100-4 x^{2}}} \mathrm{~d} x\) (m) \(\int x \cos \frac{x}{2} \mathrm{~d} x\) (n) \(\int 3^{x} \mathrm{~d} x\) (o) \(\int \sec t \mathrm{~d} t\) (p) \(\int \sin 5 x \cos 2 x \mathrm{~d} x\) (q) \(\int \frac{1}{x(x-3)} \mathrm{d} x\) (r) \(\int \frac{1}{t^{2}(t-1)} \mathrm{d} t\) (s) \(\int \frac{\ln x}{x} \mathrm{~d} x\) (t) \(\int t \sqrt{3 t-2} \mathrm{~d} t\) (u) \(\int \frac{1}{5+4 \cos x} \mathrm{~d} x\) (v) \(\int_{0}^{2} t^{2} \mathrm{e}^{t} \mathrm{~d} t\) (w) \(\int \cos x \sqrt{\sin x} \mathrm{~d} x\) (x) \(\int \frac{x^{2}}{x^{3}+1} \mathrm{~d} x\) (y) \(\int \frac{8 t+10}{4 t^{2}+8 t+3} \mathrm{~d} t\) (z) \(\int \frac{7}{\left(\mathrm{e}^{t}\right)^{7}} \mathrm{~d} t\)
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