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Second Year Math Integration Evaluate the following integrals.10. \int \frac{2 x-1}{x(x-1)(x-3)} d x


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Evaluate the following integrals.10. \int \frac{2 x-1}{x(x-1)(x-3)} d x

Evaluate the following integrals.4. \int \frac{(a-b) x}{(x-a)(x-b)} d x (a>b)
Evaluate the following integrals.4.  \int \frac{(a-b) x}{(x-a)(x-b)} d x (a>b)

Evaluate the following integrals.4. \int \frac{(a-b) x}{(x-a)(x-b)} d x (a>b)

Example 10: Evaluate(i) \int \frac{1}{\sqrt{a^{2}-x^{2}}} d x (-a<x<a) (ii) \int \frac{1}{x \sqrt{x^{2}-a^{2}}} d x(x>a or x<-a) where a is positive.
Example 10: Evaluate(i)  \int \frac{1}{\sqrt{a^{2}-x^{2}}} d x (-a<x<a) (ii)  \int \frac{1}{x \sqrt{x^{2}-a^{2}}} d x(x>a  or  x<-a) where  a  is positive.

Example 10: Evaluate(i) \int \frac{1}{\sqrt{a^{2}-x^{2}}} d x (-a<x<a) (ii) \int \frac{1}{x \sqrt{x^{2}-a^{2}}} d x(x>a or x<-a) where a is positive.

5. Evaluate the following integrals.(viii) \int \frac{e^{m \operatorname{Tan}^{-1} x}}{\left(1+x^{2}\right)} d x
5. Evaluate the following integrals.(viii)  \int \frac{e^{m \operatorname{Tan}^{-1} x}}{\left(1+x^{2}\right)} d x

5. Evaluate the following integrals.(viii) \int \frac{e^{m \operatorname{Tan}^{-1} x}}{\left(1+x^{2}\right)} d x

22. In a culture bacteria increases at the rate proportional to the number of bacteria present. If bacteria are 200 initially and are doubled in 2 hours find the number of bacteria present four hours later.
22. In a culture bacteria increases at the rate proportional to the number of bacteria present. If bacteria are 200 initially and are doubled in 2 hours find the number of bacteria present four hours later.

22. In a culture bacteria increases at the rate proportional to the number of bacteria present. If bacteria are 200 initially and are doubled in 2 hours find the number of bacteria present four hours later.

Evaluate the following integrals.12. \int \frac{4+7 x}{(1+x)^{2}(2+3 x)} d x
Evaluate the following integrals.12.  \int \frac{4+7 x}{(1+x)^{2}(2+3 x)} d x

Evaluate the following integrals.12. \int \frac{4+7 x}{(1+x)^{2}(2+3 x)} d x

Solve the following differential equations:13. \sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0
Solve the following differential equations:13.  \sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0

Solve the following differential equations:13. \sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0

Evaluate the following integrals.22. \int \frac{12}{+8} d x
Evaluate the following integrals.22.  \int \frac{12}{+8} d x

Evaluate the following integrals.22. \int \frac{12}{+8} d x

Evaluate the following integrals:15. \int \frac{\cos x}{\sin x \ln \sin x} d x
Evaluate the following integrals:15.  \int \frac{\cos x}{\sin x \ln \sin x} d x

Evaluate the following integrals:15. \int \frac{\cos x}{\sin x \ln \sin x} d x

1. Evaluate the following integrals by parts add a word representing all the functions are defined.(ii) \int \ln x d x
1. Evaluate the following integrals by parts add a word representing all the functions are defined.(ii)   \int \ln x d x

1. Evaluate the following integrals by parts add a word representing all the functions are defined.(ii) \int \ln x d x

1. Evaluate the following integrals by parts add a word representing all the functions are defined.\[\text { (xvii) } \int x \cos ^{2} x d x\]
1. Evaluate the following integrals by parts add a word representing all the functions are defined.\[\text { (xvii) } \int x \cos ^{2} x d x\]

1. Evaluate the following integrals by parts add a word representing all the functions are defined.\[\text { (xvii) } \int x \cos ^{2} x d x\]

Evaluate the following integrals.10. \int \frac{2 x-1}{x(x-1)(x-3)} d x
Evaluate the following integrals.10.  \int \frac{2 x-1}{x(x-1)(x-3)} d x
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Evaluate the following integrals.10. \int \frac{2 x-1}{x(x-1)(x-3)} d x

Example 5:Evaluate (i) \int \operatorname{cosec} x d x (ii) \int \sec x d x
Example 5:Evaluate (i)  \int \operatorname{cosec} x d x (ii)  \int \sec x d x

Example 5:Evaluate (i) \int \operatorname{cosec} x d x (ii) \int \sec x d x

Example 9: If \int_{-2}^{1} f(x) d x=5 \int_{1}^{3} f(x)=3 and \int_{-2}^{1} g(x) d x=4 then evaluate th e following definite integrals:(i) \int_{-3}^{3} f(x) d x (ii) \int_{-2}^{1}[2 f(x)+3 g(x)] d x (iii) \int_{-2}^{1} 3 f(x) d x-\int_{-2}^{1} 2 g(x) d x
Example 9: If  \int_{-2}^{1} f(x) d x=5 \int_{1}^{3} f(x)=3  and  \int_{-2}^{1} g(x) d x=4  then evaluate th e following definite integrals:(i)  \int_{-3}^{3} f(x) d x (ii)  \int_{-2}^{1}[2 f(x)+3 g(x)] d x (iii)  \int_{-2}^{1} 3 f(x) d x-\int_{-2}^{1} 2 g(x) d x

Example 9: If \int_{-2}^{1} f(x) d x=5 \int_{1}^{3} f(x)=3 and \int_{-2}^{1} g(x) d x=4 then evaluate th e following definite integrals:(i) \int_{-3}^{3} f(x) d x (ii) \int_{-2}^{1}[2 f(x)+3 g(x)] d x (iii) \int_{-2}^{1} 3 f(x) d x-\int_{-2}^{1} 2 g(x) d x

Solve the following differential equations:11. \frac{d y}{d x}+\frac{2 x y}{2 y+1}=x
Solve the following differential equations:11.  \frac{d y}{d x}+\frac{2 x y}{2 y+1}=x

Solve the following differential equations:11. \frac{d y}{d x}+\frac{2 x y}{2 y+1}=x

2. Evaluate(iii) \int \frac{d x}{\sqrt{x+a}+\sqrt{x}}(x>0 a>0)
2. Evaluate(iii)  \int \frac{d x}{\sqrt{x+a}+\sqrt{x}}(x>0 a>0)

2. Evaluate(iii) \int \frac{d x}{\sqrt{x+a}+\sqrt{x}}(x>0 a>0)

Example 1: Solve the differential equation (x-1) d x+y d y=0
Example 1: Solve the differential equation  (x-1) d x+y d y=0

Example 1: Solve the differential equation (x-1) d x+y d y=0

Evaluate the following integrals:13. \int \frac{a x}{\sqrt{a^{2}-x^{4}}}
Evaluate the following integrals:13.  \int \frac{a x}{\sqrt{a^{2}-x^{4}}}

Evaluate the following integrals:13. \int \frac{a x}{\sqrt{a^{2}-x^{4}}}

Example 7: Evaluate \int \frac{2 x}{x^{6}-1} d x
Example 7: Evaluate  \int \frac{2 x}{x^{6}-1} d x

Example 7: Evaluate \int \frac{2 x}{x^{6}-1} d x

Example 9: If \int_{-2}^{1} f(x) d x=5 \int_{1}^{3} f(x)=3 and \int_{-2}^{1} g(x) d x=4 then evaluate th e following definite integrals:(i) \int_{-3}^{3} f(x) d x (ii) \int_{-2}^{1}[2 f(x)+3 g(x)] d x (iii) \int_{-2}^{1} 3 f(x) d x-\int_{-2}^{1} 2 g(x) d x
Example 9: If  \int_{-2}^{1} f(x) d x=5 \int_{1}^{3} f(x)=3  and  \int_{-2}^{1} g(x) d x=4  then evaluate th e following definite integrals:(i)  \int_{-3}^{3} f(x) d x (ii)  \int_{-2}^{1}[2 f(x)+3 g(x)] d x (iii)  \int_{-2}^{1} 3 f(x) d x-\int_{-2}^{1} 2 g(x) d x
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Example 9: If \int_{-2}^{1} f(x) d x=5 \int_{1}^{3} f(x)=3 and \int_{-2}^{1} g(x) d x=4 then evaluate th e following definite integrals:(i) \int_{-3}^{3} f(x) d x (ii) \int_{-2}^{1}[2 f(x)+3 g(x)] d x (iii) \int_{-2}^{1} 3 f(x) d x-\int_{-2}^{1} 2 g(x) d x

Example 1 . Find \int x \cos x d x
Example  1 . Find  \int x \cos x d x
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Example 1 . Find \int x \cos x d x

Example 9: Find \int a^{x^{2}} x d x (a>0 a \neq 1)
Example 9:    Find  \int a^{x^{2}} x d x (a>0 a \neq 1)
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Example 9: Find \int a^{x^{2}} x d x (a>0 a \neq 1)

3. Use differentials to approximate the values of(iv) \sin 61^{\circ}
3. Use differentials to approximate the values of(iv)  \sin 61^{\circ}
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3. Use differentials to approximate the values of(iv) \sin 61^{\circ}

3. Find the area below the curve y=3 \sqrt{x} and above the x -axis between x=1 and x=4 .
3. Find the area below the curve  y=3 \sqrt{x}  and above the  x -axis between  x=1  and  x=4 .
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3. Find the area below the curve y=3 \sqrt{x} and above the x -axis between x=1 and x=4 .

Example 5 . Evaluate \int \ln \left(x+\sqrt{x^{2}+1}\right) d x
Example  5 . Evaluate  \int \ln \left(x+\sqrt{x^{2}+1}\right) d x
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Example 5 . Evaluate \int \ln \left(x+\sqrt{x^{2}+1}\right) d x

Evaluate the following definite integrals.1. \int_{1}^{2}\left(x^{2}+1\right) d x 2. \int_{-1}^{1}\left(x^{1 / 3}+1\right) d x 3. \int_{-2}^{0} \frac{1}{(2 x-1)^{2}} d x 4. \int_{-6}^{2} \sqrt{3-x} d x 5. \int^{\sqrt{ }} \sqrt{(2 t 1)} d t 6. \int_{2}^{\sqrt{5}} x \sqrt{x^{2}-1} d x 7. \int_{1}^{2} \frac{x}{x^{2}+2} d x 8. \int_{2}^{3}\left(x-\frac{1}{x}\right)^{2} d x 9. \int_{-1}^{1}\left(x+\frac{1}{2}\right) \sqrt{x^{2}+x+1} d x 10. \int_{0}^{3} \frac{d x}{x^{2}+9} 11. \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos t d t 12. \int_{1}^{2}\left(x+\frac{1}{x}\right)^{\frac{1}{2}}\left(1-\frac{1}{x^{2}}\right) d x 13. \int_{1}^{2} \operatorname{In} x d x 14. \int_{0}^{2}\left(e^{\frac{x}{2}}-e^{-\frac{x}{2}}\right) d x 15. \int_{0}^{\frac{\pi}{4}} \frac{\cos \theta+\sin \theta}{2 \cos ^{2} \theta} d \theta 16. \int_{0}^{\frac{\pi}{6}} \cos ^{3} \theta d \theta 17. \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \cos ^{2} \theta \cot ^{2} \theta d \theta 18. \int_{0}^{\frac{\pi}{4}} \cos ^{4} t d t
Evaluate the following definite integrals.1.  \int_{1}^{2}\left(x^{2}+1\right) d x 2.  \int_{-1}^{1}\left(x^{1 / 3}+1\right) d x 3.  \int_{-2}^{0} \frac{1}{(2 x-1)^{2}} d x 4.  \int_{-6}^{2} \sqrt{3-x} d x 5.  \int^{\sqrt{ }} \sqrt{(2 t  1)} d t 6.  \int_{2}^{\sqrt{5}} x \sqrt{x^{2}-1} d x 7.  \int_{1}^{2} \frac{x}{x^{2}+2} d x 8.  \int_{2}^{3}\left(x-\frac{1}{x}\right)^{2} d x 9.  \int_{-1}^{1}\left(x+\frac{1}{2}\right) \sqrt{x^{2}+x+1} d x 10.  \int_{0}^{3} \frac{d x}{x^{2}+9} 11.  \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos t d t 12.  \int_{1}^{2}\left(x+\frac{1}{x}\right)^{\frac{1}{2}}\left(1-\frac{1}{x^{2}}\right) d x 13.  \int_{1}^{2} \operatorname{In} x d x 14.  \int_{0}^{2}\left(e^{\frac{x}{2}}-e^{-\frac{x}{2}}\right) d x 15.  \int_{0}^{\frac{\pi}{4}} \frac{\cos \theta+\sin \theta}{2 \cos ^{2} \theta} d \theta 16.  \int_{0}^{\frac{\pi}{6}} \cos ^{3} \theta d \theta 17.  \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \cos ^{2} \theta \cot ^{2} \theta d \theta 18.  \int_{0}^{\frac{\pi}{4}} \cos ^{4} t d t
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Evaluate the following definite integrals.1. \int_{1}^{2}\left(x^{2}+1\right) d x 2. \int_{-1}^{1}\left(x^{1 / 3}+1\right) d x 3. \int_{-2}^{0} \frac{1}{(2 x-1)^{2}} d x 4. \int_{-6}^{2} \sqrt{3-x} d x 5. \int^{\sqrt{ }} \sqrt{(2 t 1)} d t 6. \int_{2}^{\sqrt{5}} x \sqrt{x^{2}-1} d x 7. \int_{1}^{2} \frac{x}{x^{2}+2} d x 8. \int_{2}^{3}\left(x-\frac{1}{x}\right)^{2} d x 9. \int_{-1}^{1}\left(x+\frac{1}{2}\right) \sqrt{x^{2}+x+1} d x 10. \int_{0}^{3} \frac{d x}{x^{2}+9} 11. \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos t d t 12. \int_{1}^{2}\left(x+\frac{1}{x}\right)^{\frac{1}{2}}\left(1-\frac{1}{x^{2}}\right) d x 13. \int_{1}^{2} \operatorname{In} x d x 14. \int_{0}^{2}\left(e^{\frac{x}{2}}-e^{-\frac{x}{2}}\right) d x 15. \int_{0}^{\frac{\pi}{4}} \frac{\cos \theta+\sin \theta}{2 \cos ^{2} \theta} d \theta 16. \int_{0}^{\frac{\pi}{6}} \cos ^{3} \theta d \theta 17. \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \cos ^{2} \theta \cot ^{2} \theta d \theta 18. \int_{0}^{\frac{\pi}{4}} \cos ^{4} t d t

1. Evaluate the following indefinite integrals(ix) \int \frac{(\sqrt{\theta}-1)^{2}}{\sqrt{\theta}} d \theta(\theta>0)
1. Evaluate the following indefinite integrals(ix)  \int \frac{(\sqrt{\theta}-1)^{2}}{\sqrt{\theta}} d \theta(\theta>0)
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1. Evaluate the following indefinite integrals(ix) \int \frac{(\sqrt{\theta}-1)^{2}}{\sqrt{\theta}} d \theta(\theta>0)

Example 4:Solve \frac{d y}{d x}=\frac{y^{2}+1}{e^{-x}}
Example 4:Solve  \frac{d y}{d x}=\frac{y^{2}+1}{e^{-x}}
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Example 4:Solve \frac{d y}{d x}=\frac{y^{2}+1}{e^{-x}}

2. Evaluate the following integral.(vii) \int e^{2 x} \cos 3 x d x
2. Evaluate the following integral.(vii)  \int e^{2 x} \cos 3 x d x
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2. Evaluate the following integral.(vii) \int e^{2 x} \cos 3 x d x

8. Find the area bounded by the curve y=x^{3}-4 x and the x -axis.
8. Find the area bounded by the curve  y=x^{3}-4 x  and the  x -axis.
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8. Find the area bounded by the curve y=x^{3}-4 x and the x -axis.

MDCAT/ ECAT question bank