Evaluate the following integrals. Write your final answers with simplified numerical coefficients and with variables raised to positive constant exponents. If your answer is a radical expression, no need to convert the expression with rational exponents. Show complete solution and/or substitution for full credit. 1. ?(5y^4 + 4y^3 - 3y^2 + 2y - 8)dy 2. ? ?(x)(x + 1/x) dx 3. ? t(2t - 3?(t)) dt 4. ? (tan^2 x + cot^2 x + 4) dx 5. ? (sin x + cos x / sin^2 x) dx 6. ? cos x / (1 + sin x)^5 dx 7. ? 1/x^3 ((4x^2 - 1) / x^2)^(2/3) dx 8. ? (?(x) - 1)^6 dx 9. ? x csc^3 x^2 cot x^2 dx 10. ? (2?(3x^2 - 1) - 2/x) (3x / ?(3x^2 - 1) + 1/x^2) dx 11. ? ln x dx / x 12. ? dx / (x + ?(x)) HINT: Factor the denominator 13. ? x^3 dx / (x^4 - 2x^2 + 1) HINT: Factor the denominator 14. ?_0^3 x dx / (4x^2 + 2) 15. ? 5?(x) e^(5?(x)) / ?(x) dx 16. ? e^(x + e^x) dx Recall properties of exponents 17. ? dx / (x ln x ?((ln x)^2 - 8)) 18. ? dx / (9x^2 + 25) 19. ? (8x + 1) / (4x^2 + 12x + 10) dx 20. ? dx / ?(7 + 6x - x^2)
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$\int \frac{1}{x\sqrt{x+1}} dx$ Let $u = x + 1$, then $du = dx$ and $x = u - 1$. The integral becomes: $\int \frac{1}{(u-1)\sqrt{u}} du$ Now let $v = \sqrt{u}$, then $dv = \frac{1}{2\sqrt{u}} du$ and $u = v^2$. The integral becomes: $2\int \frac{1}{(v^2-1)v} Show more…
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