Please do help solve the followings
1) Integrate e^4 dx
2) Integrate dx/sqrt(90^2-4x^2)
3) Integrate (e^x+x)^2(e^x+1) dx
4) Integrate xe^x2 dx
e^4 is a constant.
3) let u= e^x + x
du= (e^x + 1)dx
4) let u= x du=dx
v= e^x dv= e^x dx
Sure, let's solve these integrals step by step.
1) To integrate e^4 dx, we can simply treat e^4 as a constant because it is not dependent on x. Therefore, the integral becomes e^4 * x + C, where C is the constant of integration.
2) To integrate dx/sqrt(90^2-4x^2), we can use the trigonometric substitution technique. Let's substitute x = (90/2)sin(theta). Then, dx = (90/2)cos(theta) dtheta, and sqrt(90^2-4x^2) becomes 90cos(theta). Substituting these values into the integral, the integral becomes:
∫ (90/2)cos(theta) [(90/2)cos(theta)] dtheta
= (90/2)^2 ∫ cos^2(theta) dtheta
We can then use the trigonometric identity cos^2(theta) = (1+cos(2theta))/2 to simplify the integral:
(90/2)^2 ∫ (1+cos(2theta))/2 dtheta
= (90/2)^2 * (1/2) ∫ (1+cos(2theta)) dtheta
= (90/2)^2 * (1/2) (theta + (1/2)sin(2theta))
Finally, we substitute theta back with its original value x = (90/2)sin(theta) to obtain the final integral expression.
3) To integrate (e^x+x)^2(e^x+1) dx, we can use a substitution to simplify the expression. Let u = e^x + x, then du = (e^x + 1) dx. Rewriting the integral in terms of u, it becomes:
∫ u^2 du
= (1/3)u^3 + C
Substituting back u = e^x + x will give us the final integral expression.
4) To integrate xe^(x^2) dx, we can use the technique of integration by parts. Let's choose u = x and dv = e^(x^2) dx. Then, du = dx and v = (1/2)e^(x^2).
Using the formula for integration by parts: ∫ u dv = uv - ∫ v du, the integral becomes:
∫ xe^(x^2) dx = (1/2)x * e^(x^2) - ∫ (1/2)e^(x^2) dx
The second term on the right side can be easily evaluated by substitution, letting u = x^2 and du = 2x dx. The integral becomes:
∫ (1/2)e^(x^2) dx = (1/2) ∫ e^u du = (1/2) * e^u + C = (1/2) * e^(x^2) + C
Substituting this back into the initial integral expression gives the final result:
∫ xe^(x^2) dx = (1/2)x * e^(x^2) - (1/2) * e^(x^2) + C.