Question

5.5 u - substitution

1. find the following indefinite integrals.

a. \\(\int (3x + 5)^7 dx\\)
f. \\(\int \frac{x^3}{x^2 + 3} dx\\)
b. \\(\int \frac{x}{\sqrt{1 - x^2}} dx\\)
g. \\(\int \sin\theta \cos\theta(1 - \cos\theta)^{20} d\theta\\)
c. \\(\int t(1 - t^2)^5 d\\) (note: there may be a typo here, likely should be (dt))
h. \\(\int t\sqrt{5 + t} dt\\)
d. \\(\int \cos^5 x \sin x dx\\)
i. \\(\int x^5\sqrt{1 + x^2} dx\\)
e. \\(\int 7\sin(\tan(w)) \sec^2(w) dw\\)
j. \\(\int e^{2x}(1 + e^x)^3 dx\\)

Explanation:

Step1: Let \(u = 3x+5\)

Differentiate \(u\) with respect to \(x\): \(du=3dx\), so \(dx=\frac{1}{3}du\)

Step2: Substitute into the integral

\(\int(3x + 5)^{7}dx=\int u^{7}\cdot\frac{1}{3}du\)

Step3: Integrate

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq- 1\)), we have \(\frac{1}{3}\int u^{7}du=\frac{1}{3}\cdot\frac{u^{8}}{8}+C\)

Step4: Back - substitute

Since \(u = 3x+5\), the integral is \(\frac{(3x + 5)^{8}}{24}+C\)

Answer:

\(\frac{(3x + 5)^{8}}{24}+C\)