Use the substitution u = tan x to evaluate the integral
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Question

Use the substitution u = tan x to evaluate the integral

∫ dx / (1 + sin² x).

Accepted Answer

Welcome back everyone. In this problem, we want to evaluate the integral of dx divided by 2 + sin square x using the substitution u equals 10 x. Now, let's first see how we can use our substitution to help us replace dx and sin square x with terms involving you. So if we let you, so by substitution, if we let you. Be equal to the tangent of x. Then that means X. OK, let me put this here. That means X. Will be equal to the inverse tangent of you. Now if we differentiate this, this means that the derivative of x with respect to u is going to be equal to 1 divided by 1 + u2, which means that dx will be equal to DU divided by 1 + u2. So actually to make some space, I could remove this and just take here dx to be equal to DU. Divided by 1 + U2, OK. Now we have DD X in terms of you. How about the sin squared of X? Well, recall, OK, that. Sin squared of x is really times 2 x divided by sex square of x. So this is how we can relate our tangent to our sine term. And now if we take it even further, remember that sex squared x is really 1+ times square x. Since 10 x is u, that means that 12 x is u 2, so our answer our sin squared x will really be u2 divided by 1 plus u2. Now that we have sin squared x and dx in terms of u, this means then that our integral of d x divided by 2 plus sin square x. Will be equal to the integral. OK, of 1 divided by 2 + u2 divided by 1 + U2 multiplied by DU divided by 1 + U2. Now we might be able to make our problems simpler by simplifying our denominator. Now if we take the denominator here, 2 + u2 divided by 1 + u2. Then if we write it as one term, it will be 2 multiplied by 1 + u2 + u2 divided by 1 + u2. Which would simplify to 2 + 2U2 + U2 or 2 + 3 U 2 divided by 1 + u2 and if we put that back into our problem. OK, then that means it's going to become the integral of 1 + u2 divided by 2 + 3 u 2 because remember we're going to reciprocate that first term in our product multiplied by dU divided by 1 plus u 2 and now you notice that 1 + u2 cancels. To give us the integral of DU divided by 2 + 3 U 2. Now we could simplify this even further, OK, because if we think about it, we can factor the coefficient of u 2. So really we could rewrite this as 1/3 multiplied by the integral of duu divided by 2/3, OK, because remember we're dividing through by 3, we're factoring out 1/3, OK. Plus U2, OK, cause 3U2 divided by 3 would leave us with U2. No, look at our integral here. Does it look familiar? Have you ever seen any other integral in this form? Well, recall, eh. That the integral of 1 divided by a 2 plus u2 with respect to u would be equal to 1 divided by a multiplied by the inverse tangent of u divided by a plus c. and in this case if we think about a squad as 2/3, then a would be equal to the square root of 2/3. So what does that mean for us? Well, if we apply this formula. This tells us then, OK, that 1/3 of our integral of duu divided by 2/3 a 2 plus u2 will be equal. To 1, 1/3, yeah. Multiplied by 1 divided by a, so 1 divided by the square root of 2/3, OK, multiplied by the inverse tangent of u divided by a, the square root of 2/3 plus c. And know if we simplify this. OK, then it's going to be 1/3 multiplied by the square root of 3 halves multiplied by the inverse tangent of u multiplied by the square root of 3 halves plus c and we take it a step further. Then this is going to be the square root of 6 divided by 6 multiplied by the inverse tangent of the square root of 6 divided by 2 multiplied by u which we know that u is. And x, so it's really the square root of 6 multiplied by the tangent of x divided by 2. Let me write that back. Let me write that a little better, a little clearer, OK. So the square root of 6 multiplied by the tangent of x divided by 2 plus c. Therefore this would be the integral. OK, of d x divided by 2 plus sin squared x if we were to use the substitution that u equals the tangent of x. Thanks a lot for watching everyone. I hope this video helped.

Use the substitution u = tan x to evaluate the integral∫ dx / (1 + sin² x).

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Use the substitution u = tan x to evaluate the integral
∫ dx / (1 + sin² x).