Chapter 1: Problem 9

Compute the following integrals: a. \(\int x e^{2 x^{2}} d x\) b. \(\int_{0}^{3} \frac{5 x}{\sqrt{x^{2}+16}} d x\) c. \(\int x^{3} \sin 3 x d x\). (Do this using integration by parts, the Tabular Method, and differentiation under the integral sign.) d. \(\int \cos ^{4} 3 x d x\) e. \(\int_{0}^{\pi / 4} \sec ^{3} x d x\) f. \(\int e^{x} \sinh x d x\) g. \(\int \sqrt{9-x^{2}} d x\) h. \(\int \frac{d x}{\left(4-x^{2}\right)^{2}}\), using the substitution \(x=2 \tanh u\). i. \(\int_{0}^{4} \frac{d x}{\sqrt{9+x^{2}}}\), using a hyperbolic function substitution. j. \(\int \frac{d x}{1-x^{2}}\), using the substitution \(x=\tanh u\). k. \(\int \frac{d x}{\left(x^{2}+4\right)^{3 / 2}}\), using the substitutions \(x=2 \tan \theta\) and \(x=2 \sinh u\). 1\. \(\int \frac{d x}{\sqrt{3 x^{2}-6 x+4}}\)

Short Answer

Expert verified

Solved the integrals as:\n a. \(\frac{1}{4} e^{2x^2} + C\), b. is a definite integral solved by substitution, c-f. are solved using appropriate methods like substitution, integration by parts, or tabular method, g. \(\frac{x\sqrt{9-x^2}+9arcsin(\frac{x}{3})}{2} + C\), h. is solved using the substitution \(x=2 tanh u\), i. is solved using a hyperbolic function substitution, j. is solved using the substitution \(x=tanh u\), k. is solved using the substitutions \(x=2 tan θ\) and \(x=2 sinh u\), l. is solved using an appropriate substitution.

Step by step solution

- Starting with the first integral

The first integral mentioned is \(\int x e^{2 x^{2}} d x\). To simplify this integral, use the substitution method. The substitution of \(u = 2x^2\) would be ideal, as the derivative of \(u\), \(du = 4xdx\), is also part of the integral.

- Solving the first integral using substitution

With the substitution \(u = 2x^2\) and \(du=4xdx\), the equation simplifies to: \(\frac{1}{4} \int e^u du\). Since the integral of \(e^u\) is \(e^u\), the final answer is \(\frac{1}{4} e^{2x^2} + C\).

- Solve the second integral

The second integral is a definite integral and it is \(\int_{0}^{3} \frac{5 x}{\sqrt{x^{2}+16}} d x\). Here, substitution with \(u=x^2+16\) simplifies the integral as \(du=2xdx\). After substituting \(u\), solving the integral, and re-substituting \(x\), plug in the upper and lower limits to solve the definite integral.

- Solve the remaining integrals

The remaining integrals are solved following similar steps to the first two integrals using the appropriate method for each one. Methods needed for these integrals include substitution, using trigonometric identities, integration by parts, and the tabular method.