
Solve .
Answer
513.3k+ views
Hint:Here we assume the whole integral as a variable and the solve by integration by parts where using the sequence of ILATE ( Inverse trigonometric, Logarithms , algebraic, trigonometric , exponential) for choosing the value of and then substitute in the integration formula of by parts integration.
* If we have then the value of integral is given by where is differentiation of with respect to .
Complete step-by-step answer:
Let us assume the value of whole integral as
i.e.
Now, using the ILATE sequence we can tell that the term comes first in the sequence is and which comes later is i.e. in this case we have the terms exponential and trigonometric , first comes the trigonometric term then comes the exponential term so is and is
Now we calculate the differentiation of the term
Now substitute the values in equation
Now we know where is constant.
Therefore substitute the value of in equation
Now we can take out constant term in the denominator of integration out of the integration
Now we can see there is again an integration left on RHS of the equation, therefore we solve it separately and substitute it back in the equation
To solve
We use using the ILATE sequence we can tell that the term comes first in the sequence is and which comes later is i.e. in this case we have the terms exponential and trigonometric , first comes the trigonometric term then comes the exponential term so is and is
Now we calculate the differentiation of the term
Now substitute the values in equation
Now we know where is constant.
Therefore substitute the value of in equation
Now we can take out constant term in the denominator of integration out of the integration
From equation we know
Therefore substitute value of in equation
Now we substitute back the value of in equation
{ is multiplied to bracket and }
Now take LCM of the terms in RHS of the equation
Shifting the term with to the LHS of the equation.
Taking LCM on LHS of the equation
Now we can cancel out the denominator on both sides of the equations because they are equal.
Take common on RHS of the equation
Now to find the value of we divide both sides of the equation by
Therefore, value of the integral
Note:Students are likely to make mistake in integration by parts when they don’t do it step by step, that is instead of substituting the value of differentiation after it is done separately they differentiate the term after substituting it into the formula which causes confusion sometimes because the second half of the integration has double integration.
* If we have
Complete step-by-step answer:
Let us assume the value of whole integral as
i.e.
Now, using the ILATE sequence we can tell that the term comes first in the sequence is
Now we calculate the differentiation of the term
Now substitute the values in equation
Now we know
Therefore substitute the value of
Now we can take out constant term in the denominator of integration out of the integration
Now we can see there is again an integration left on RHS of the equation, therefore we solve it separately and substitute it back in the equation
To solve
We use using the ILATE sequence we can tell that the term comes first in the sequence is
Now we calculate the differentiation of the term
Now substitute the values in equation
Now we know
Therefore substitute the value of
Now we can take out constant term in the denominator of integration out of the integration
From equation
Therefore substitute value of
Now we substitute back the value of
Now take LCM of the terms in RHS of the equation
Shifting the term with
Taking LCM on LHS of the equation
Now we can cancel out the denominator on both sides of the equations because they are equal.
Take
Now to find the value of
Therefore, value of the integral
Note:Students are likely to make mistake in integration by parts when they don’t do it step by step, that is instead of substituting the value of differentiation after it is done separately they differentiate the term after substituting it into the formula which causes confusion sometimes because the second half of the integration has double integration.
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