Question

5.5 u-substitution

1. find the following indefinite integrals.

a. \\( \int (3x + 5)^7 dx \\)

b. \\( \int \frac{x}{\sqrt{1 - x^2}} dx \\)

c. \\( \int t(1 - t^2)^5 d \\)

d. \\( \int \cos^5 x \\ \sin x \\, dx \\)

e. \\( \int 7 \sin(\tan(w)) \sec^2(w) \\, dw \\)

Explanation:

Step1: Solve $\int(3x + 5)^{7}dx$

Let \(u = 3x+5\), then \(du=3dx\), \(dx=\frac{1}{3}du\).

$$
LATEXBLOCK0
$$

Step2: Solve $\int\frac{x}{\sqrt{1 - x^{2}}}dx$

Let \(u = 1-x^{2}\), then \(du=-2xdx\), \(xdx=-\frac{1}{2}du\).

$$
LATEXBLOCK1
$$

Step3: Solve $\int t(1 - t^{2})^{5}dt$

Let \(u = 1 - t^{2}\), then \(du=-2tdt\), \(tdt=-\frac{1}{2}du\).

$$
LATEXBLOCK2
$$

Step4: Solve $\int\cos^{5}x\sin xdx$

Let \(u=\cos x\), then \(du=-\sin xdx\), \(\sin xdx=-du\).

$$
LATEXBLOCK3
$$

Step5: Solve $\int7\sin(\tan(w))\sec^{2}(w)dw$

Let \(u = \tan w\), then \(du=\sec^{2}wdw\).

$$
LATEXBLOCK4
$$

Answer:

A. \(\frac{(3x + 5)^{8}}{24}+C\)
B. \(-\sqrt{1 - x^{2}}+C\)
C. \(-\frac{(1 - t^{2})^{6}}{12}+C\)
D. \(-\frac{\cos^{6}x}{6}+C\)
E. \(-7\cos(\tan w)+C\)