Lectures>Practice

# Exercises

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## Exercises

Evaluate the following integrals.

 $\int 4x^3(x^4 + 1)^4 dx$ $\int 2x^2(x^3 + 5)^5 dx$ $\int (x^2 + 2x)(x^3 + 3x^2 + 1)^5 dx$ $\int \cos(3\theta) d\theta$ $\int x^2 \sqrt{x^3 + 2} dx$ $\int \frac{1}{3x - 4} dx$ $\int \frac{x}{\sqrt{3x^2 + 4}} dx$ $\int \left(\sin \theta \cos^{5} \theta\right) d\theta$ $\int \frac{\sec^2 t}{1 + \tan t} dt$ $\int xe^{x^2} dx$ $\int 6^{4x} dx$ $\int \frac{2}{x\sqrt{x^2 + 1}} dx$ $\int \frac{x}{x + 3} dx$ $\int x^5 \cos(x^6) dx$ $\int \frac{2x^3 + 3x}{x^4 + 3x^2} dx$ $\int \left((z + 1)\sqrt{z^2 + 2z}\right) dz$ $\int \left(\cos^2 x\right) dx$ $\int 3^{x - 2} dx$ $\int \frac{\sin x}{\cos^2 x} dx$ $\int \sin^2(2x) dx$ $\int x\cos(x^2) dx$ $\int \sec 4x \tan 4x dx$ $\int \frac{\cos 2x}{\sin 2x - 1} dx$ $\int \frac{e^x}{e^x + 1} dx$ $\int\sin^2 x\cos^2 x dx$

## Explorations

Evaluate the following integrals.

 $\int \cos(x + a) dx$ $\int \sin(x + a) dx$ $\int \frac{1}{x + a} dx$
1. Suppose $\int f(x) dx = F(x)$. What's $\int f(x + a)dx$?
 $\int \cos(nx) dx$ $\int \sqrt{nx} dx$ $\int e^{nx} dx$
1. Suppose $\int f(x) dx = F(x)$. What's $\int f(nx)dx$?
1. Show that $\int\left(f(x)\right)^n f'(x) dx = \frac{1}{n+1}\left(f(x)\right)^{n+1} + C, n \ne -1$.

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