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4.1 E Exercises

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Exercise $$\PageIndex{1}$$

1. Why is u-substitution referred to as change of variable?

2. If $$\displaystyle f=g∘h$$, when reversing the chain rule, $$\displaystyle \frac{d}{d}x(g∘h)(x)=g′(h(x))h′(x)$$, should you take $$\displaystyle u=g(x)$$ or u= $$\displaystyle h(x)?$$

$$\displaystyle u=h(x)$$.

Exercise $$\PageIndex{2}$$

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $$\displaystyle∫f(u)du.$$

1. $$\displaystyle∫x\sqrt{x+1}x=\frac{2}{15}(x+1)^{3/2}(3x−2)+C;u=x+1$$

2. $$\displaystyle∫\frac{x^2}{\sqrt{x−1}}dx(x>1)=\frac{2}{15}\sqrt{x−1}(3x^2+4x+8)+C;u=x−1$$

$$\displaystyle f(u)=\frac{(u+1)^2}{\sqrt{u}}$$

3. $$\displaystyle∫x\sqrt{4x^2+9}dx=\frac{1}{12}(4x^2+9)^{3/2}+C;u=4x^2+9$$

4. $$\displaystyle∫\frac{x}{\sqrt{4x^2+9}}dx=\frac{1}{4}\sqrt{4x^2+9}+C;u=4x^2+9$$

$$\displaystyle du=8xdx;f(u)=\frac{1}{8\sqrt{u}}$$

5. $$\displaystyle∫\frac{x}{(4x^2+9)^2}dx=−\frac{1}{8(4x^2+9)};u=4x^2+9$$

$$\displaystyle du=8xdx;f(u)=\frac{1}{8u^2}$$

Exercise $$\PageIndex{3}$$

In the following exercises, find the antiderivative using the indicated substitution.

1. $$\displaystyle ∫(x+1)^4dx;u=x+1$$

$$\displaystyle \frac{1}{5}(x+1)^5+C$$

2. $$\displaystyle∫(x−1)^5dx;u=x−1$$

3. $$\displaystyle∫(2x−3)^{−7}dx;u=2x−3$$

$$\displaystyle−\frac{1}{12(3−2x)^6}+C$$

4. $$\displaystyle∫(3x−2)^{−11}dx;u=3x−2$$

5. $$\displaystyle∫\frac{x}{\sqrt{x^2+1}}dx;u=x^2+1$$

$$\displaystyle\sqrt{x^2+1}+C$$

6. $$\displaystyle∫\frac{x}{\sqrt{1−x^2}}dx;u=1−x^2$$

7. $$\displaystyle∫(x−1)(x^2−2x)^3dx;u=x^2−2x$$

$$\displaystyle\frac{1}{8}(x^2−2x)^4+C$$

8. $$\displaystyle∫(x^2−2x)(x^3−3x^2)^2dx;u=x^3=3x^2$$

9. $$\displaystyle∫cos^3θdθ;u=sinθ (Hint:cos^2θ=1−sin^2θ)$$

$$\displaystylesinθ−\frac{sin^3θ}{3}+C$$

10. $$\displaystyle∫sin^3θdθ;u=cosθ (Hint:sin^2θ=1−cos^2θ)$$

Exercise $$\PageIndex{4}$$

In the following exercises, use a suitable change of variables to determine the indefinite integral.

1. $$\displaystyle∫x(1−x)^{99}dx$$

$$\displaystyle\frac{(1−x)^{101}}{101}−\frac{(1−x)^{100}}{100}+C$$

2. $$\displaystyle∫t(1−t^2)^{10}dt$$

3. $$\displaystyle∫(11x−7)^{−3}dx$$

$$\displaystyle−\frac{1}{22(7−11x^2)}+C$$

4. \displaystyle∫(7x−11)^4dx\)

5. $$\displaystyle∫cos^3θsinθdθ$$

$$\displaystyle−\frac{cos^4θ}{4}+C$$

6. $$\displaystyle∫sin^7θcosθdθ$$

7. $$\displaystyle∫cos^2(πt)sin(πt)dt$$

$$\displaystyle−\frac{cos^3(πt)}{3π}+C$$

8. $$\displaystyle∫sin^2xcos^3xdx (Hint:sin^2x+cos^2x=1)$$

9. $$\displaystyle∫tsin(t^2)cos(t^2)dt$$

$$\displaystyle−\frac{1}{4}cos^2(t^2)+C$$

10. $$\displaystyle∫t^2cos^2(t^3)sin(t^3)dt$$

11. $$\displaystyle∫\frac{x^2}{(x^3−3)^2}dx$$

$$\displaystyle−\frac{1}{3(x^3−3)}+C$$

12. $$\displaystyle∫\frac{x^3}{\sqrt{1−x^2}}dx$$

13. $$\displaystyle∫\frac{y^5}{(1−y^3)^{3/2}}dy$$

$$\displaystyle−\frac{2(y^3−2)}{3\sqrt{1−y^3}}$$

14. $$\displaystyle∫cosθ(1−cosθ)^{99}sinθdθ$$

15. $$\displaystyle∫(1−cos^3θ)^{10}cos^2θsinθdθ$$

$$\displaystyle\frac{1}{33}(1−cos^3θ)^{11}+C$$

16. $$\displaystyle∫(cosθ−1)(cos^2θ−2cosθ)^3sinθdθ$$

17. $$\displaystyle∫(sin^2θ−2sinθ)(sin^3θ−3sin^2θ^)3cosθdθ$$

$$\displaystyle\frac{1}{12}(sin^3θ−3sin^2θ)^4+C$$

Exercise $$\PageIndex{5}$$

1) $$\displaystyle \int \frac{ x^3\ dx}{\sqrt{1-x^2}}$$

2) $$\displaystyle\int (\csc x)(\cot x)(e^{\csc x})\,\,dx$$

3)$$\displaystyle \int \frac{1}{e^x+e^{-x}}dx$$

4) $$\displaystyle \int \frac {1}{\sqrt{4-x^2}} \ dx$$

5) $$\displaystyle \int \frac{\sqrt{x^2-4}}{x} \ dx$$

6) $$\displaystyle \int \frac{1}{x^2 + 6x + 13} dx$$