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Section 7.3 : Trig Substitutions

For problems 1 – 8 use a trig substitution to eliminate the root.

1. \(\sqrt {4 - 9{z^2}} \) Solution
2. \(\sqrt {13 + 25{x^2}} \) Solution
3. \({\left( {7{t^2} - 3} \right)^{\frac{5}{2}}}\) Solution
4. \(\sqrt {{{\left( {w + 3} \right)}^2} - 100} \) Solution
5. \(\sqrt {4{{\left( {9t - 5} \right)}^2} + 1} \) Solution
6. \(\sqrt {1 - 4z - 2{z^2}} \) Solution
7. \({\left( {{x^2} - 8x + 21} \right)^{\frac{3}{2}}}\) Solution
8. \(\sqrt {{{\bf{e}}^{8x}} - 9} \) Solution

For problems 9 – 16 use a trig substitution to evaluate the given integral.

1. \( \displaystyle \int{{\frac{{\sqrt {{x^2} + 16} }}{{{x^4}}}\,dx}}\) Solution
2. \( \displaystyle \int{{\sqrt {1 - 7{w^2}} \,dw}}\) Solution
3. \( \displaystyle \int{{{t^3}{{\left( {3{t^2} - 4} \right)}^{\frac{5}{2}}}\,dt}}\) Solution
4. \( \displaystyle \int_{{ - 7}}^{{ - 5}}{{\frac{2}{{{y^4}\sqrt {{y^2} - 25} }}\,dy}}\) Solution
5. \( \displaystyle \int_{1}^{4}{{2{z^5}\sqrt {2 + 9{z^2}} \,dz}}\) Solution
6. \( \displaystyle \int{{\frac{1}{{\sqrt {9{x^2} - 36x + 37} }}\,dx}}\) Solution
7. \( \displaystyle \int{{\frac{{{{\left( {z + 3} \right)}^5}}}{{{{\left( {40 - 6z - {z^2}} \right)}^{\frac{3}{2}}}}}\,dz}}\) Solution
8. \( \displaystyle \int{{\cos \left( x \right)\sqrt {9 + 25 \sin^2\left( x \right)} \,dx}}\) Solution