1 e 37 pi sin pi log ex x dx is equal to 525021830

$$\int_1^{{e^{37}}} {\frac{{\pi \sin \left( {\pi {{\log }_e}x} \right)}}{x}dx} $$ is equal to -

A. $$2$$

B. $$-2$$

C. $$\frac{2}{\pi }$$

D. $$2\pi $$

Answer : $$2$$

Solution :
$$\eqalign{ & {\text{Put}}\,\,{\log _e}x = z \cr & {\text{Then }}I = \int_0^{37} {\pi .\sin \,\pi z\,dz} \cr & = \left[ { - \cos \,\pi z} \right]_0^{37} \cr & = - \cos \,37\pi + \cos \,0 \cr & = 1 + 1 \cr & = 2 \cr} $$

Releted MCQ Question on
Calculus >> Definite Integration

Releted Question 1

The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,dx$$ is-

A. $$ - 1$$

B. $$2$$

C. $$1 + {e^{ - 1}}$$

D. none of these

View Answer

Releted Question 2

Let $$a,\,b,\,c$$ be non-zero real numbers such that $$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = } \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has-

A. no root in $$\left( {0,\,2} \right)$$

B. at least one root in $$\left( {0,\,2} \right)$$

C. a double root in $$\left( {0,\,2} \right)$$

D. two imaginary roots

View Answer

Releted Question 3

The value of the integral $$\int\limits_0^{\frac{\pi }{2}} {\frac{{\sqrt {\cot \,x} }}{{\sqrt {\cot \,x} + \sqrt {\tan \,x} }}dx} $$ is-

A. $$\frac{\pi }{4}$$

B. $$\frac{\pi }{2}$$

C. $$\pi $$

D. none of these

View Answer

Releted Question 4

For any integer $$n$$ the integral $$\int\limits_0^\pi {{e^{{{\cos }^2}x}}} {\cos ^3}\left( {2n + 1} \right)xdx$$ has the value-

A. $$\pi $$

B. $$1$$

C. $$0$$

D. none of these

View Answer

$$\int_1^{{e^{37}}} {\frac{{\pi \sin \left( {\pi {{\log }_e}x} \right)}}{x}dx} $$ is equal to -

Releted MCQ Question on
Calculus >> Definite Integration

The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,dx$$ is-

Let $$a,\,b,\,c$$ be non-zero real numbers such that $$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = } \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has-

The value of the integral $$\int\limits_0^{\frac{\pi }{2}} {\frac{{\sqrt {\cot \,x} }}{{\sqrt {\cot \,x} + \sqrt {\tan \,x} }}dx} $$ is-

For any integer $$n$$ the integral $$\int\limits_0^\pi {{e^{{{\cos }^2}x}}} {\cos ^3}\left( {2n + 1} \right)xdx$$ has the value-

Practice More Releted MCQ Question on
Definite Integration

Practice More MCQ Question on Maths Section

$$\int_1^{{e^{37}}} {\frac{{\pi \sin \left( {\pi {{\log }_e}x} \right)}}{x}dx} $$ is equal to -

Releted MCQ Question on Calculus >> Definite Integration

The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,dx$$ is-

Let $$a,\,b,\,c$$ be non-zero real numbers such that $$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = } \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} $$ Then the quadratic equation $$a{x^2} + bx + c = 0$$ has-

The value of the integral $$\int\limits_0^{\frac{\pi }{2}} {\frac{{\sqrt {\cot \,x} }}{{\sqrt {\cot \,x} + \sqrt {\tan \,x} }}dx} $$ is-

For any integer $$n$$ the integral $$\int\limits_0^\pi {{e^{{{\cos }^2}x}}} {\cos ^3}\left( {2n + 1} \right)xdx$$ has the value-

Practice More Releted MCQ Question on Definite Integration

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Calculus >> Definite Integration

Let $$a,\,b,\,c$$ be non-zero real numbers such that $$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = } \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has-

Practice More Releted MCQ Question on
Definite Integration