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# 4.7 E Exercises

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Exercise $$\PageIndex{1}$$

In the following exercises, use a change of variables to evaluate the definite integral.

1)  $$\displaystyle∫^1_0x\sqrt{1−x^2}dx$$

2)  $$\displaystyle∫^1_0\frac{x}{\sqrt{1+x^2}}dx$$

Answer

$$\displaystyleu=1+x^2,du=2xdx,\frac{1}{2}∫^2_1u^{−1/2}du=\sqrt{2}−1$$

3) $$\displaystyle∫^2_0\frac{t}{\sqrt{5+t^2}}dt$$

4) $$\displaystyle∫^1_0\frac{t}{\sqrt{1+t^3}}dt$$

Answer:

$$\displaystyleu=1+t^3,du=3t^2,\frac{1}{3}∫^2_1u^{−1/2}du=\frac{2}{3}(\sqrt{2}−1)$$

5) $$\displaystyle∫^{π/4}_0sec^2θtanθdθ$$

6) $$\displaystyle∫^{π/4}_0\frac{sinθ}{cos^4θ}dθ$$

Answer:

$$\displaystyleu=cosθ,du=−sinθdθ,∫^1_{1/\sqrt{2}}u^{−4}du=\frac{1}{3}(2\sqrt{2}−1)$$

Exercise $$\PageIndex{2}$$

In the following exercises, evaluate the definite integral.

1) $$\displaystyle ∫^2_1\frac{1+2x+x^2}{3x+3x^2+x^3}dx$$

Answer

$$\displaystyle \frac{1}{3}ln(\frac{26}{7})$$

2) $$\displaystyle ∫^{π/4}_0tanxdx$$

3) $$\displaystyle ∫^{π/3}_0\frac{sinx−cosx}{sinx+cosx}dx$$

Answer:

$$\displaystyle ln(\sqrt{3}−1)$$

4)  $$\displaystyle ∫^{π/2}_{π/6}cscxdx$$

5) $$\displaystyle ∫^{π/3}_{π/4}cotxdx$$

Answer:

$$\displaystyle \frac{1}{2}ln\frac{3}{2}$$

Exercise $$\PageIndex{3}$$

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

1. $$\displaystyle∫^π_0cos^2(2θ)sin(2θ)dθ$$
2.  $$\displaystyle∫^\sqrt{π}_0tcos(t^2)sin(t^2)dt$$

Answer:

$$\displaystyle u=sin(t^2);$$ the integral becomes $$\displaystyle\frac{1}{2}∫^0_0udu.$$

3.  $$\displaystyle∫^1_0(1−2t)dt$$

4.  $$\displaystyle∫^1_0\frac{1−2t}{(1+(t−\frac{1}{2})^2)}dt$$

Answer:

$$\displaystyleu=(1+(t−\frac{1}{2})^2);$$ the integral becomes $$\displaystyle−∫^{5/4}_{5/4}\frac{1}{u}du$$.

5. $$\displaystyle∫^π_0sin((t−\frac{π}{2})^3)cos(t−\frac{π}{2})dt$$

6. $$\displaystyle∫^2_0(1−t)cos(πt)dt$$

Answer

$$\displaystyle∫^{−1}_1ucos(π(1−u))du=∫^{−1}_1u[cosπcosu−sinπsinu]du=−∫^{−1}_1ucosudu=∫^{1−}_1ucosudu=0$$

since the integrand is odd.

7.  $$\displaystyle∫^{3π/4}_{π/4}sin^2tcostdt$$

Exercise $$\PageIndex{4}$$

In the following exercises, evaluate the definite integral.

Answer

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