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Chapter 6: Problem 6

Use a table of integrals with forms involving the trigonometric functions to find the integral. $$ \int \frac{\cos ^{3} \sqrt{x}}{\sqrt{x}} d x $$

Short Answer

Expert verified
The integral of \(\int \frac{\cos (\sqrt{x})^{3}}{\sqrt{x}} dx\) equals to \(\frac{2}{3}\sin(\sqrt{x}^{3}) + C\).

Step by step solution

01

Substitution

Use the substitution \(u = \sqrt{x}\), or equivalently \(x = u^2\), which implies that \(dx = 2udu\). Now replace \(x\) and \(dx\) in the original integral with the expressions involving \(u\).
02

Simplify Integral

After substitution, the integral becomes \(\int 2u \cos(u^{3})du\). Notice that the integral is now in the form of \(cos(u^3)\times u\), which is equivalent to the derivative of \(\frac{1}{3} \sin(u^3)\). This observation allows the use of the standard integral form: \(\int cos(nu)du = \frac{1}{n} sin(nu) + C\), where \(n = u^2\).
03

Final Calculation

Applying the formula, we obtain \(\int 2u \cos(u^{3})du = \frac{2}{3}\sin(u^{3}) + C\). Now replace the \(u\) by \(\sqrt{x}\) that was the initial substitution. The final answer is \(\frac{2}{3}\sin(\sqrt{x}^{3}) + C\).

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Most popular questions from this chapter

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \cos ^{4} \frac{x}{2} d x $$

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\cot ^{2} t}{\csc t} d t $$

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\cosh a t $$

(a) find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only by a constant. $$ \int \sec ^{2} x \tan x d x $$

(A) find the indefinite integral in two different ways. (B) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (C) Verify analytically that the results differ only by a constant. $$ \int \sec ^{4} 3 x \tan ^{3} 3 x d x $$
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